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Year: 2019

Multiple mixing and parabolic divergence in smooth area-preserving flowson higher genus surfaces

Kanigowski, Adam ; Kułaga-Przymus, Joanna ; Ulcigrai, Corinna

Abstract: We consider typical area-preserving flows on higher genus surfaces and prove that the flowrestricted to mixing minimal components is mixing of all orders, thus answering affirmatively Rokhlin’smultiple mixing question in this context. The main tool is a variation of the Ratner property origi-nally proved by Ratner for the horocycle flow, i.e. the switchable Ratner property introduced by Fayadand Kanigowski for special flows over rotations. This property, which is of independent interest, pro-vides a quantitative description of the parabolic behavior of these flows and has implications for joiningclassification. The main result is formulated in the language of special flows over interval exchange trans-formations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayadand Kanigowski’s main results, by showing that Arnold’s flows are mixing of all orders for almost everylocation of the singularities.

DOI: https://doi.org/10.4171/jems/914

Posted at the Zurich Open Repository and Archive, University of ZurichZORA URL: https://doi.org/10.5167/uzh-180479Journal ArticleAccepted Version

Originally published at:Kanigowski, Adam; Kułaga-Przymus, Joanna; Ulcigrai, Corinna (2019). Multiple mixing and parabolicdivergence in smooth area-preserving flows on higher genus surfaces. Journal of the European Mathe-matical Society, 21(12):3797-3855.DOI: https://doi.org/10.4171/jems/914

▼❯▲❚■P▲❊ ▼■❳■◆● ❆◆❉ P❆❘❆❇❖▲■❈ ❉■❱❊❘●❊◆❈❊ ■◆ ❙▼❖❖❚❍

❆❘❊❆✲P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ❖◆ ❍■●❍❊❘ ●❊◆❯❙ ❙❯❘❋❆❈❊❙✳

❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

❆❜str❛❝t✳ ❲❡ ❝♦♥s✐❞❡r t②♣✐❝❛❧ ❛r❡❛ ♣r❡s❡r✈✐♥❣ ✢♦✇s ♦♥ ❤✐❣❤❡r ❣❡♥✉s s✉r❢❛❝❡s ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡ ✢♦✇r❡str✐❝t❡❞ t♦ ♠✐①✐♥❣ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✱ t❤✉s ❛♥s✇❡r✐♥❣ ❛✣♠❛t✐✈❡❧② t♦ ❘♦❤❧✐♥✬s♠✉❧t✐♣❧❡ ♠✐①✐♥❣ q✉❡st✐♦♥ ✐♥ t❤✐s ❝♦♥t❡①t✳ ❚❤❡ ♠❛✐♥ t♦♦❧ ✐s ❛ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ♦r✐❣✐♥❛❧❧②♣r♦✈❡❞ ❜② ❘❛t♥❡r ❢♦r t❤❡ ❤♦r♦❝②❝❧❡ ✢♦✇✱ ✐✳❡✳ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ✐♥tr♦❞✉❝❡❞ ❜② ❋❛②❛❞ ❛♥❞❑❛♥✐❣♦✇s❦✐ ❢♦r s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r r♦t❛t✐♦♥s✳ ❚❤✐s ♣r♦♣❡rt②✱ ✇❤✐❝❤ ✐s ♦❢ ✐♥❞❡♣❡♥❞❡♥t ✐♥t❡r❡st✱ ♣r♦✈✐❞❡s❛ q✉❛♥t✐t❛t✐✈❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♣❛r❛❜♦❧✐❝ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡s❡ ✢♦✇s ❛♥❞ ❤❛s ✐♠♣❧✐❝❛t✐♦♥ t♦ ❥♦✐♥✐♥❣s❝❧❛ss✐✜❝❛t✐♦♥✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ✐s ❢♦r♠✉❧❛t❡❞ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ❲❡ ❛❧s♦ ♣r♦✈❡ ❛ str❡♥❣t❤❡♥✐♥❣ ♦❢ ♦♥❡ ♦❢❋❛②❛❞ ❛♥❞ ❑❛♥✐❣♦✇s❦✐✬s ♠❛✐♥ r❡s✉❧ts✱ ❜② s❤♦✇✐♥❣ t❤❛t ❆r♥♦❧❞✬s ✢♦✇s ❛r❡ ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦❞❡rs ❢♦r ❛❧♠♦st❡✈❡r② ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s✳

❈♦♥t❡♥ts

✶✳ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠❛✐♥ r❡s✉❧ts✳ ✷✶✳✶✳ ▲♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✷✶✳✷✳ ▼✐①✐♥❣✱ ❘♦❦❤❧✐♥✬s q✉❡st✐♦♥ ❛♥❞ ♠✉❧t✐♣❧❡ ♠✐①✐♥❣ ✸✶✳✸✳ P❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ❛♥❞ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ✹✶✳✹✳ ❖✉t❧✐♥❡ ❛♥❞ str✉❝t✉r❡ ♦❢ t❤❡ ♣❛♣❡r ✻✷✳ ❇❛❝❦❣r♦✉♥❞ ♠❛t❡r✐❛❧ ✽✷✳✶✳ ▲♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✽✷✳✷✳ ❙♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s ✾✷✳✸✳ ▲♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ❛s s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s ✶✵✷✳✹✳ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ♦❢ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ✶✶✷✳✺✳ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥ ✶✷✷✳✻✳ ❘❛✉③②✲❱❡❡❝❤ ❛❝❝❡❧❡r❛t✐♦♥s ✶✺✷✳✼✳ P♦s✐t✐✈✐t②✱ ❜❛❧❛♥❝❡✱ ♣r❡✲❝♦♠♣❛❝t♥❡ss ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ t❛✐❧s ✶✻✸✳ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ❢♦r ■❊❚s ✶✼✸✳✶✳ ▼✐①✐♥❣ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✶✽✸✳✷✳ ❘❛t♥❡r ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✶✽✸✳✸✳ ◗✉❛s✐✲❇❡r♥♦✉❧❧✐ ♣r♦♣❡rt② ✷✵✸✳✹✳ ❋✉❧❧ ♠❡❛s✉r❡ ♦❢ t❤❡ ❘❛t♥❡r ❉❈ ✷✸✹✳ ❇✐r❦❤♦✛ s✉♠s ♦❢ r♦♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ✷✺✹✳✶✳ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ❢♦r s♣❡❝✐❛❧ ✢♦✇s ✭♦✈❡r ■❊❚s✮ ✈✐❛ ❇✐r❦❤♦✛ s✉♠s ✷✺✹✳✷✳ ●r♦✇t❤ ♦❢ ❇✐r❦❤♦✛ s✉♠s ♦❢ ❞❡r✐✈❛t✐✈❡s ✷✻✹✳✸✳ ❘❛t♥❡r ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✷✾✺✳ Pr♦♦❢ ♦❢ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ✸✵✺✳✶✳ ❈♦♥tr♦❧ ♦❢ ❡✐t❤❡r ❜❛❝❦✇❛r❞ ♦r ❢♦r✇❛r❞ ♦r❜✐ts ❞✐st❛♥❝❡ ❢r♦♠ s✐♥❣✉❧❛r✐t✐❡s ✸✶✺✳✷✳ Pr♦♦❢ t❤❛t t❤❡ ❘❛t♥❡r ❉❈ ✐♠♣❧✐❡s t❤❡ ❙❘✲♣r♦♣❡rt② ✸✺✺✳✸✳ ❈♦♥❝❧✉s✐♦♥s ✸✾❆♣♣❡♥❞✐① ❆✳ ✹✵❆✳✶✳ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ❛r❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t ✹✵❆✳✷✳ ❙✐♥❣✉❧❛r✐t✐❡s ❞✐st❛♥❝❡s ❝♦♥tr♦❧ ❜② ♣♦s✐t✐✈❡ ❘❛✉③②✲❱❡❡❝❤ t✐♠❡s ✹✶❆❝❦♥♦✇❧❡❞❣♠❡♥ts ✹✷❘❡❢❡r❡♥❝❡s ✹✷

✶

✷ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

✶✳ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠❛✐♥ r❡s✉❧ts✳

■♥ t❤✐s ♣❛♣❡r ✇❡ ❣✐✈❡ ❛ ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ ❡r❣♦❞✐❝ t❤❡♦r② ♦❢ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇s ❛♥❞✱ ♠♦r❡ ✐♥ ❣❡♥❡r❛❧t♦ t❤❡ st✉❞② ♦❢ ♣❛r❛❜♦❧✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠s✳ ❙✐♥❝❡ t❤❡ ♦r✐❣✐♥s ♦❢ t❤❡ st✉❞② ♦❢ ❞②♥❛♠✐❝s✱ ✇✐t❤ P♦✐♥❝❛ré✱✢♦✇s ♦♥ s✉r❢❛❝❡s ❤❛✈❡ ❜❡❡♥ ♦♥❡ ♦❢ t❤❡ ❜❛s✐❝ ❡①❛♠♣❧❡s ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s✳ ❲❡ ❝♦♥s✐❞❡r s♠♦♦t❤✢♦✇s ✇❤✐❝❤ ♣r❡s❡r✈❡ ❛ s♠♦♦t❤ ❛r❡❛ ❢♦r♠✱ ❛❧s♦ ❦♥♦✇♥ ❛s ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✶✮✳ ■♥t❤✐s ❝♦♥t❡①t✱ ✇❡ ❛❞❞r❡ss ❘♦❦❤❧✐♥ q✉❡st✐♦♥ ♦♥ ♠✉❧t✐♣❧❡ ♠✐①✐♥❣ ✭s❡❡ ❙❡❝t✐♦♥ ✶✳✷✮ ❛♥❞ ♣r♦✈❡ ❛ ✈❡rs✐♦♥ ♦❢❘❛t♥❡r✬s ♣r♦♣❡rt② ♦♥ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ✭✭s❡❡ ❙❡❝t✐♦♥ ✶✳✸✮✳

✶✳✶✳ ▲♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s✳ ❉❡♥♦t❡ ❜② S ❛ s♠♦♦t❤ ❝❧♦s❡❞ ❝♦♥♥❡❝t❡❞ ♦r✐❡♥t❛❜❧❡ s✉r❢❛❝❡ ♦❢ ❣❡♥✉sg ≥ 1✱ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ st❛♥❞❛r❞ ❛r❡❛ ❢♦r♠ ω ✭♦❜t❛✐♥❡❞ ❛s ♣✉❧❧✲❜❛❝❦ ♦❢ t❤❡ ❛r❡❛ ❢♦r♠ dxdy ♦♥R2✮✳ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r ❛ s♠♦♦t❤ ✢♦✇ (ϕt)t∈R ♦♥ S ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛ ♠❡❛s✉r❡ µ ❣✐✈❡♥ ✐♥t❡❣r❛t✐♥❣ ❛

s♠♦♦t❤ ❞❡♥s✐t② ✇✐t❤ r❡s♣❡❝t t♦ ω✳ ❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡ ❛r❡❛ ✐s ♥♦r♠❛❧✐③❡❞ s♦ t❤❛t µ(S) = 1✳ ❆s❡①♣❧❛✐♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✶✱ s♠♦♦t❤ ❛r❡❛ ♣r❡s❡r✈✐♥❣ ✢♦✇s ❛r❡ ✐♥ ♦♥❡ t♦ ♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ s♠♦♦t❤❝❧♦s❡❞ r❡❛❧✲✈❛❧✉❡❞ ❞✐✛❡r❡♥t✐❛❧ 1✲❢♦r♠s ❛♥❞ ❛r❡ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s✱ ❛❧s♦ ❦♥♦✇♥ ❛s ♠✉❧t✐✲✈❛❧✉❡❞❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s✳ ❆ ❧♦t ♦❢ ✐♥t❡r❡st ✐♥ t❤❡ st✉❞② ♦❢ ♠✉❧t✐✲✈❛❧✉❡❞ ❍❛♠✐❧t♦♥✐❛♥s ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ✢♦✇s✕ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐♥ t❤❡✐r ❡r❣♦❞✐❝ ❛♥❞ ♠✐①✐♥❣ ♣r♦♣❡rt✐❡s ✕ ✇❛s s♣❛r❦❡❞ ❜② ◆♦✈✐❦♦✈ ❬✸✷❪ ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤♣r♦❜❧❡♠s ❛r✐s✐♥❣ ✐♥ s♦❧✐❞✲st❛t❡ ♣❤②s✐❝s ✭✐✳❡✳ t❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ❡❧❡❝tr♦♥ ✐♥ ❛ ♠❡t❛❧ ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ ❛♠❛❣♥❡t✐❝ ✜❡❧❞✮ ❛♥❞ ✐♥ ♣s❡✉❞♦✲♣❡r✐♦❞✐❝ t♦♣♦❧♦❣② ✭s❡❡ ❡✳❣✳ t❤❡ s✉r✈❡② ❜② ❩♦r✐❝❤ ❬✺✵❪✮✳

❲❤❡♥ g ≥ 2✱ t❤❡ ✭✜♥✐t❡✮ s❡t ♦❢ ✜①❡❞ ♣♦✐♥ts ♦❢ (ϕt)t∈R ✐s ❛❧✇❛②s ♥♦♥✲❡♠♣t②✳ ❆ ❣❡♥❡r✐❝ ❧♦❝❛❧❧②❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❇❛✐r❡ ❝❛t❡❣♦r②✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t♦♣♦❧♦❣② ❣✐✈❡♥ ❜② ❝♦♥s✐❞❡r✐♥❣♣❡rt✉r❜❛t✐♦♥s ♦❢ ❝❧♦s❡❞ s♠♦♦t❤ 1✲❢♦r♠s ❜② ✭s♠❛❧❧✮ ❝❧♦s❡❞ s♠♦♦t❤ 1✲❢♦r♠s✮ ❤❛s ♦♥❧② ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞♣♦✐♥ts✱ ✐✳❡✳ ❝❡♥t❡rs ✭s❡❡ ❋✐❣✉r❡ ✶✭❛✮✮ ❛♥❞ s✐♠♣❧❡ s❛❞❞❧❡s ✭s❡❡ ❋✐❣✉r❡ ✶✭❜✮✮✱ ❛s ♦♣♣♦s❡❞ t♦ ❞❡❣❡♥❡r❛t❡♠✉❧t✐✲s❛❞❞❧❡s ✇❤✐❝❤ ❤❛✈❡ 2k s❡♣❛r❛tr✐①❡s ❢♦r k > 2 ✭s❡❡ ❋✐❣✉r❡ ✶✭❝✮✮✳ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝s ✭❛s ♣r♦✈❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❜② ▼❛✐❡r ❬✷✾❪✱ ▲❡✈✐tt ❬✷✺❪ ❛♥❞ ❩♦r✐❝❤ ❬✺✵❪✮✱ ❡❛❝❤ s♠♦♦t❤❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇ ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥ts ❛♥❞ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts✿ ❛ ♣❡r✐♦❞✐❝❝♦♠♣♦♥❡♥t ✐s ❛ s✉❜s✉r❢❛❝❡ ✭♣♦ss✐❜❧② ✇✐t❤ ❜♦✉♥❞❛r②✮ ♦♥ ✇❤✐❝❤ ❛❧❧ ♦r❜✐ts ❛r❡ ❝❧♦s❡❞ ❛♥❞ ♣❡r✐♦❞✐❝ ✭s❡❡❢♦r ❡①❛♠♣❧❡ ❋✐❣✉r❡ ✶✭❛✮ ♦r ❋✐❣✉r❡ ✶✭❞✮✮❀ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ✭t❤❡r❡ ❛r❡ ♥♦t ♠♦r❡ t❤❛♥ g ♦❢ t❤❡♠✮ ❛r❡s✉❜s✉r❢❛❝❡s ✭♣♦ss✐❜❧② ✇✐t❤ ❜♦✉♥❞❛r②✮ ♦♥ ✇❤✐❝❤ t❤❡ ✢♦✇ ✐s ♠✐♥✐♠❛❧ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❛❧❧ s❡♠✐✲✐♥✜♥✐t❡tr❛❥❡❝t♦r✐❡s ❛r❡ ❞❡♥s❡ ✭s❡❡ ❋✐❣✉r❡ ✷✮✳

✭❛✮ ✭❜✮ ✭❝✮ ✭❞✮

❋✐❣✉r❡ ✶✳ ❚②♣❡ ♦❢ ✜①❡❞ ♣♦✐♥ts ❛♥❞ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥ts ✐♥ ❛♥ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇✳

❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t❤❡ ❡r❣♦❞✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❛ t②♣✐❝❛❧ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ✐♥ t❤❡ s❡♥s❡ ♦❢ ♠❡❛s✉r❡✲t❤❡♦r②✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡ ❛ ♠❡❛s✉r❡ ❝❧❛ss ♦♥ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✭s♦♠❡t✐♠❡s ❝❛❧❧❡❞❑❛t♦❦ ❢✉♥❞❛♠❡♥t❛❧ ❝❧❛ss✱ s❡❡ ❙❡❝t✐♦♥ ✷✳✶ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥✮✿ ✇❤❡♥ ✇❡ s❛② t②♣✐❝❛❧✱ ✇❡ ♠❡❛♥ ❢✉❧❧ ♠❡❛s✉r❡✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ♠❡❛s✉r❡ ❝❧❛ss✳ ❖♥❡ ❝❛♥ ❞✐✈✐❞❡ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✐♥t♦ t✇♦ ♦♣❡♥ s❡ts ✭✐♥ t❤❡t♦♣♦❧♦❣② ❣✐✈❡♥ ❜② ♣❡rt✉r❜❛t✐♦♥s ❜② ❝❧♦s❡❞ s♠♦♦t❤ 1✲❢♦r♠s✱ s❡❡ ❙❡❝t✐♦♥ ✷✳✶ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✮✿ ✐♥ t❤❡ ✜rst♦♣❡♥ s❡t✱ ✇❤✐❝❤ ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② Umin✱ t❤❡ t②♣✐❝❛❧ ✢♦✇ ✐s ♠✐♥✐♠❛❧ ✭✐♥ ♣❛rt✐❝✉❧❛r t❤❡r❡ ❛r❡ ♥♦ ❝❡♥t❡rs❛♥❞ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t✮ ❛♥❞ ❡r❣♦❞✐❝ ✭✐✳❡✳ t❤❡r❡ ❛r❡ ♥♦ ♠❡❛s✉r❡❛❜❧❡ ✢♦✇✲✐♥✈❛r✐❛♥ts❡ts A ⊂ S s✉❝❤ t❤❛t µ(A) /∈ {0, 1}✮✳ ❖♥ t❤❡ ♦t❤❡r ♦♣❡♥ s❡t t❤❛t ✇❡ ✇✐❧❧ ❝❛❧❧ U¬min t❤❡r❡ ❛r❡ ♣❡r✐♦❞✐❝❝♦♠♣♦♥❡♥ts ✭❜♦✉♥❞❡❞ ❜② s❛❞❞❧❡ ❧♦♦♣s ❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦✮✱ ❜✉t ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ t②♣✐❝❛❧ ✢♦✇s❛r❡ st✐❧❧ ♠✐♥✐♠❛❧ ❛♥❞ ✉♥✐q✉❡❧② ❡r❣♦❞✐❝✳ ❇♦t❤ r❡s✉❧ts ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠✐♥✐♠❛❧❝♦♠♣♦♥❡♥ts ❛s s♣❡❝✐❛❧ ✢♦✇s ❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✷ ❜❡❧♦✇ ❛♥❞ t❤❡ ❝❧❛ss✐❝❛❧ r❡s✉❧ts ❜② ❑❡❛♥❡ ❬✶✼❪ ❛♥❞

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✸

❋✐❣✉r❡ ✷✳ ❉❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥ts ✜❧❧❡❞ ❜② ❝❧♦s❡❞ ♦r❜✐ts ✭t✇♦ ✐s❧❛♥❞s ❛r♦✉♥❞❝❡♥t❡rs ❛♥❞ ❛ ❝②❧✐♥❞❡r ✐♥ ❜❧✉❡ ✐♥ t❤❡ ❋✐❣✉r❡✮ ❛♥❞ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ✭♦♥❡ ♦❢ ♦❢ ❣❡♥✉s ♦♥❡ ❛♥❞♦♥❡ ♦❢ ❣❡♥✉s t✇♦ ✐♥ t❤❡ ❡①❛♠♣❧❡✮✳

▼❛s✉r ❛♥❞ ❱❡❡❝❤ ✭s❡❡ ❬✷✽✱ ✹✺❪✮ r❡s♣❡❝t✐✈❡❧② ❝♦♥❝❡r♥✐♥❣ ♦❢ ♠✐♥✐♠❛❧✐t② ❛♥❞ ❡r❣♦❞✐❝✐t② ♦❢ t②♣✐❝❛❧ ✐♥t❡r✈❛❧❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✳

✶✳✷✳ ▼✐①✐♥❣✱ ❘♦❦❤❧✐♥✬s q✉❡st✐♦♥ ❛♥❞ ♠✉❧t✐♣❧❡ ♠✐①✐♥❣✳ ❙tr♦♥❣❡r ❝❤❛♦t✐❝ ♣r♦♣❡rt✐❡s t❤❛♥ ❡r❣♦❞✐❝✐t②❛r❡ ♠✐①✐♥❣ ❛♥❞ ♠✉❧t✐♣❧❡ ✭♦r ❤✐❣❤❡r ♦r❞❡r✮ ♠✐①✐♥❣✳ ❆ ✢♦✇ (ϕt)t∈R ♣r❡s❡r✈✐♥❣ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ✐s ♠✐①✐♥❣ ✭♦r str♦♥❣❧② ♠✐①✐♥❣✮ ✐❢ ❛♥② ♣❛✐r ♦❢ ♠❡❛s✉r❛❜❧❡ s❡ts A,B ❜❡❝♦♠❡ ❛s②♠♣t♦t✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✉♥❞❡r t❤❡ ✢♦✇✱ ✐✳❡✳ limt→∞ µ(ϕt(A)∩B) = µ(A)µ(B)✳ ▼♦r❡ ✐♥ ❣❡♥❡r❛❧✱ (ϕt)t∈R ✐s ♠✐①✐♥❣ ♦❢ ♦r❞❡r k ≥ 2✐❢ ❢♦r ❛♥② k ♠❡❛s✉r❛❜❧❡ s❡ts A1, A2, . . . , Ak✱

limt2,...,tk→∞

µ(A1 ∩ ϕt2(A2) ∩ ϕt2+t3(A3) · · · ∩ ϕt2+···+tk(Ak) = µ(A1)µ(A2) . . . µ(Ak).

❈❧❡❛r❧②✱ ❢♦r k = 2 t❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥ r❡❞✉❝❡s t♦ ♠✐①✐♥❣✳ ■❢ ❛ ✢♦✇ ✐s ♠✐①✐♥❣ ♦❢ ♦r❞❡r k ❢♦r ❛♥② k ≥ 2✱✇❡ s❛② t❤❛t ✐t ✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✳

❆r♥♦❧❞ ✐♥ ❬✶❪ ❝♦♥❥❡❝t✉r❡❞ t❤❛t ✇❤❡♥ g = 1 ❛♥❞ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥t ✭✐✳❡✳ ♦♥❡ ✐s ✐♥U¬min✮✱ t❤❡ t②♣✐❝❛❧ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ r❡str✐❝t❡❞ t♦ t❤❡ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ✐s ♠✐①✐♥❣✳ ❋♦❧❧♦✇✐♥❣❬✶✵❪✱ ✇❡ ✇✐❧❧ ❝❛❧❧ ❆r♥♦❧❞ ✢♦✇ t❤❡ r❡str✐❝t✐♦♥ ♦❢ s✉❝❤ ❛ ✢♦✇ t♦ ✐ts ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t✳ ❚❤✐s ❝♦♥❥❡❝t✉r❡✇❛s ♣r♦✈❡❞ ❜② ❑❤❛♥✐♥ ❛♥❞ ❙✐♥❛✐ ❬✸✾❪ ✭s❡❡ ❛❧s♦ ❢✉rt❤❡r ✇♦r❦s ❜② ❑♦❝❤❡r❣✐♥ ❬✷✶✱ ✷✷✱ ✷✸✱ ✷✹❪✮✳ ❖♥ t❤❡ ♦t❤❡r❤❛♥❞✱ ✐♥ Umin✱ ✇❤❡♥ g = 1✱ t❤❡ t②♣✐❝❛❧ ✢♦✇ ✐s ♥♦t ♠✐①✐♥❣ ✭t❤✐s ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ❡✐t❤❡r ❢r♦♠ t❤❡ ✇♦r❦❬✶✾❪ ❜② ❑♦❝❤❡r❣✐♥ ♦r ❜② ❝❧❛ss✐❝❛❧ ❑❆▼ t❤❡♦r②✮✳ ▼✐①✐♥❣ ♦♥ s✉r❢❛❝❡s ♦❢ ❤✐❣❤❡r ❣❡♥✉s ✭✐✳❡✳ ✇❤❡♥ g ≥ 2✮✇❛s ✐♥✈❡st✐❣❛t❡❞ ❜② t❤❡ ❧❛st ❛✉t❤♦r✳ ❙❤❡ s❤♦✇❡❞ ❬✹✸❪ t❤❛t ✐♥ t❤❡ ♦♣❡♥ s❡t Umin✱ t❤❡ t②♣✐❝❛❧ ✢♦✇✱ ✇❤✐❝❤✐s ♠✐♥✐♠❛❧ ❛♥❞ ❡r❣♦❞✐❝✱ ✐s ♥♦t ♠✐①✐♥❣ ✭s❡❡ ❛❧s♦ ❬✸✽❪ ❢♦r ❛♥ ✐♥❞❡♣❡♥❞❡♥t ♣r♦♦❢ ♦❢ t❤❡ s❛♠❡ r❡s✉❧t ✇❤❡♥g = 2✮✱ ❡✈❡♥ t❤♦✉❣❤ ✐t ✐s ✇❡❛❦❧② ♠✐①✐♥❣ ❬✹✷❪✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❘❛✈♦tt✐ ❬✸✺❪✱ ❜② ❣❡♥❡r❛❧✐③✐♥❣ t❤❡ ♠❛✐♥r❡s✉❧t ♣r♦✈❡❞ ❜② t❤❡ ❧❛st ❛✉t❤♦r ✐♥ ❬✹✶❪ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ s♣❡❝✐❛❧ ✢♦✇s✱ r❡❝❡♥t❧② s❤♦✇❡❞ t❤❛t t②♣✐❝❛❧ ✢♦✇s✐♥ U¬min ❤❛✈❡ ♠✐①✐♥❣ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ❛♥❞ ♣r♦✈✐❞❡❞ q✉❛♥t✐t❛t✐✈❡ ❜♦✉♥❞s ♦♥ t❤❡ s♣❡❡❞ ♦❢ ♠✐①✐♥❣❢♦r s♠♦♦t❤ ♦❜s❡r✈❛❜❧❡s ✭s❤♦✇✐♥❣ t❤❛t ♠✐①✐♥❣ ❤❛♣♣❡♥s ❛t s✉❜♣♦❧②♥♦♠✐❛❧ r❛t❡s✮✳ ▲❡t ✉s ❛❧s♦ r❡❝❛❧❧ t❤❛t✐♥ t❤❡ 1970s ❑♦❝❤❡r❣✐♥ ❬✷✵❪ ♣r♦✈❡❞ ♠✐①✐♥❣ ✇❤❡♥ t❤❡r❡ ❛r❡ ❞❡❣❡♥❡r❛t❡ s❛❞❞❧❡s ✭t❤❛t ✐s✱ ✐♥ ❛ ♥♦♥✲❣❡♥❡r✐❝❝❛s❡✮ ❛♥❞ t❤❛t ✈❡r② r❡❝❡♥t❧② ❈❤❛✐❦❛ ❛♥❞ ❲r✐❣❤t ❬✽❪ s❤♦✇❡❞ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠✐①✐♥❣ ✢♦✇s ✐♥ Umin ✭✇❤✐❝❤❜② ❬✹✸❪ ❝♦♥s✐st✉t❡ ❛ ♠❡❛s✉r❡ ③❡r♦ ❡①❝❡♣t✐♦♥❛❧ s❡t✮✳

❆ ❢❛♠♦✉s ❛♥❞ st✐❧❧ ✇✐❞❡❧② ♦♣❡♥ ♣r♦❜❧❡♠ ✐♥ ❡r❣♦❞✐❝ t❤❡♦r② ✐s t❤❡ q✉❡st✐♦♥ ❜② ❘♦❦❤❧✐♥ ✇❤❡t❤❡r ♠✐①✐♥❣✐♠♣❧✐❡s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs ❬✸✻❪✳ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇s✱ ❋❛②❛❞ ❛♥❞ t❤❡ ✜rst ❛✉t❤♦rr❡❝❡♥t❧② ♣r♦✈❡❞ ✐♥ ❬✶✵❪ t❤❛t ✇❤❡♥ g = 1 ✢♦✇s ✇✐t❤ ❛ ♠✐①✐♥❣ ♠✐♥✐♠❛❧ ❝♦♠♣♦♠❡♥t ✭❛s ✇❡❧❧ ❛s s♦♠❡♠✐①✐♥❣ ❑♦❝❤❡r❣✐♥ ✢♦✇s ✇✐t❤ ❞❡❣❡♥❡r❛t❡ s❛❞❞❧❡s✮ ❛r❡ ✐♥❞❡❡❞ ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✱ t❤✉s ✈❡r✐❢②✐♥❣ ❘♦❦❤❧✐♥✬sq✉❡st✐♦♥ ✐♥ t❤✐s ❝♦♥t❡①t✳

❖✉r ♠❛✐♥ r❡s✉❧t ✐s t❤❛t ♠✐①✐♥❣ ✐♠♣❧✐❡s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs ❢♦r t②♣✐❝❛❧ s♠♦♦t❤ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇s♦♥ s✉r❢❛❝❡s ♦❢ ❛♥② ❣❡♥✉s✳

❚❤❡♦r❡♠ ✶✳✶✳ ❋♦r ❛♥② ✜①❡❞ ❣❡♥✉s g ≥ 1✱ ❝♦♥s✐❞❡r ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ♦♥ ❛ s✉r❢❛❝❡ S ♦❢ ❣❡♥✉sg ✇✐t❤ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts ❛♥❞ ❛t ❧❡❛st ♦♥❡ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥t✳ ❋♦r ❛ t②♣✐❝❛❧ ✢♦✇ (ϕt)t∈R ✐♥ ❛♥♦♣❡♥ ❛♥❞ ❞❡♥s❡ s❡t✱ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (ϕt)t∈R t♦ ❛♥② ♦❢ ✐ts ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✳

▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ♦♣❡♥ ❛♥❞ ❞❡♥s❡ s❡t ♦❢ ✢♦✇s ✇✐t❤ ❛t ❧❡❛st ♦♥❡ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥t ✐♥ t❤❡ st❛t❡♠❡♥t✐s t❤❡ s❛♠❡ s❡t ✐♥ U¬min ❢♦r ✇❤✐❝❤ ♦♥❡ ❝❛♥ ❛❧s♦ ♣r♦✈❡ t❤❛t t②♣✐❝❛❧❧② ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ❛r❡ ♠✐①✐♥❣

✹ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

✭s❡❡ ❬✸✺❪✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ s✐♥❝❡ t②♣✐❝❛❧ ✢♦✇s ♦♥ Umin ❛r❡ ♥♦t ♠✐①✐♥❣ ❜② ❬✹✸❪✱ ✐t ❢♦❧❧♦✇s t❤❛t ❢♦r ❛ t②♣✐❝❛❧❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ♠✐①✐♥❣ ✭♦❢ ♦♥❡ ♦❢ ✐ts ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts✮ ✐♠♣❧✐❡s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✳

❚❤✐s r❡s✉❧t ✐s ❞❡❞✉❝❡❞ ❢r♦♠ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ✭s❡❡ ❚❤❡♦r❡♠ ✶✳✷❜❡❧♦✇✮✳ ❈♦♥s✐❞❡r ❛ s❡❣♠❡♥t tr❛♥s✈❡rs❡ t♦ ❛ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ❛♥ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇ ❛♥❞ t❤❡❛ss♦❝✐❛t❡❞ P♦✐♥❝❛ré ✜rst r❡t✉r♥ ♠❛♣ ✭✐✳❡✳ t❤❡ ♠❛♣ t❤❛t s❡♥❞s ❛ ♣♦✐♥t t♦ t❤❡ ✜rst ♣♦✐♥t ❛❧♦♥❣ ✐ts ✢♦✇♦r❜✐t t❤❛t ❤✐ts t❤❡ s❛♠❡ s❡❣♠❡♥t ❛❣❛✐♥✮✳ P♦✐♥❝❛ré ♠❛♣s ♦❢ s♠♦♦t❤ ❛r❡❛ ♣r❡s❡r✈✐♥❣ ✢♦✇s✱ ✐♥ s✉✐t❛❜❧❡❝♦♦r❞✐♥❛t❡s✱ ❛r❡ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s ✭❢♦r s❤♦rt✱ ■❊❚s✮✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ✐s♦♠❡tr✐❡s ♦❢❛♥ ✐♥t❡r✈❛❧ I ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✷ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥✮✳ ●✐✈❡♥ ❛♥ ■❊❚ T : I → I ✇❤✐❝❤ ♦❝❝✉rs ❛s ❛ P♦✐♥❝❛rés❡❝t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡ ❛♥❞ t❤❡ r❡t✉r♥ t✐♠❡ ❢✉♥❝t✐♦♥ ✭✇❤✐❝❤ ✐s ❛♥ ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ f : I → R ❞❡✜♥❡❞❛t ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥② ♣♦✐♥ts✮ ♦♥❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ✢♦✇ ❛s ❢♦❧❧♦✇s✳ ▲❡t Xf ❝♦♥s✐st ♦❢ ♣♦✐♥ts ❜❡❧♦✇t❤❡ ❣r❛♣❤ ♦❢ f ✱ t❤❛t ✐s Xf + {(x, y) ∈ R

2 : x ∈ I, 0 ≤ y < f(x)}✳ ❯♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ s♣❡❝✐❛❧✢♦✇ (ϕt)t∈R ♦✈❡r t❤❡ ♠❛♣ T ✉♥❞❡r t❤❡ r♦♦❢ f ❛ ♣♦✐♥t (x, y) ∈ Xf ♠♦✈❡s ✇✐t❤ ✉♥✐t ✈❡❧♦❝✐t② ❛❧♦♥❣ t❤❡✈❡rt✐❝❛❧ ❧✐♥❡ ✉♣ t♦ t❤❡ ♣♦✐♥t (x, f(x))✱ t❤❡♥ ❥✉♠♣s ✐♥st❛♥t❧② t♦ t❤❡ ♣♦✐♥t (T (x), 0)✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❜❛s❡tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ ❛❢t❡r✇❛r❞ ✐t ❝♦♥t✐♥✉❡s ✐ts ♠♦t✐♦♥ ❛❧♦♥❣ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ✉♥t✐❧ t❤❡ ♥❡①t ❥✉♠♣ ❛♥❞ s♦♦♥✳ ❚❤❡ ❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ✐s ❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✷✳✷✳ ❙✐♥❝❡ T ♣r❡s❡r✈❡s t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✱ ❛♥② s♣❡❝✐❛❧✢♦✇ ♦✈❡r T ♣r❡s❡r✈❡s t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ t♦ Xf ✳ ■t ✐s ✇❡❧❧ ❦♥♦✇♥t❤❛t t❤❡ ♦r✐❣✐♥❛❧ ✢♦✇ ♦♥ t❤❡ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ✐s ♠❡❛s✉r❡✲t❤❡♦r❡t✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❞❡s❝r✐❜❡❞s♣❡❝✐❛❧ ✢♦✇ ❛♥❞ ❤❡♥❝❡ ❤❛s t❤❡ s❛♠❡ ❡r❣♦❞✐❝ ❛♥❞ ♠✐①✐♥❣ ♣r♦♣❡rt✐❡s✳

❊❛❝❤ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ❛ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ s♣❡❝✐❛❧ ✢♦✇ ♦✈❡r❛♥ ■❊❚✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r♦♦❢ ❢✉♥❝t✐♦♥ ✐s ♥♦t ❞❡✜♥❡❞ ❛t ❛ s✉❜s❡t ♦❢ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ t❤❡ ■❊❚ ✇❤✐❝❤❝♦rr❡s♣♦♥❞ t♦ ♣♦✐♥ts t❤❛t ❤✐t ❛ s❛❞❞❧❡ ❜❡❢♦r❡ t❤❡✐r ✜rst r❡t✉r♥✱ s❡❡ ❙❡❝t✐♦♥ ✷✳✸✳ ❙✐♥❝❡ t❤❡ ✢♦✇ ✐s s♠♦♦t❤✱t❤❡s❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ❛r❡ s✐♥❣✉❧❛r✐t✐❡s ♦❢ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥ ✭✇❡ ❤❛✈❡ f(x) → +∞ ❛s x ❛♣♣r♦❝❤❡s ♦♥❡♦r ❜♦t❤ s✐❞❡s ♦❢ s✉❝❤ ❛ ❞✐s❝♦♥t✐♥✉✐t②✮✳ ■t t✉r♥s ♦✉t t❤❛t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s✐♠♣❧❡ s❛❞❞❧❡s ❣✐✈❡ r✐s❡ t♦❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ♦❢ f ✱ ✐✳❡✳ ♣♦✐♥ts ✇❤❡r❡ f ❜❧♦✇s ✉♣ ❛s t❤❡ ❢✉❝t✐♦♥ | log(x)| ♥❡❛r ③❡r♦✱ ✐♥ ❛ s❡♥s❡♠❛❞❡ ♣r❡❝✐s❡ ✐♥ ❙❡❝t✐♦♥ ✷✳✸ ✭✇❤✐❧❡ ❞❡❣❡♥❡r❛t❡ s❛❞❞❧❡s ❣✐✈❡ r✐s❡ t♦ ♣♦❧②♥♦♠✐❛❧ s✐♥❣✉❧❛r✐t✐❡s✱ ✇❤✐❝❤ ❛r❡t❤❡ t②♣❡ ♦❢ s✐♥❣✉❧❛r✐t✐❡s ❝♦♥s✐❞❡r❡❞ ❜② ❑♦❝❤❡r❣✐♥ ✐♥ ❬✷✵❪ ❛♥❞ ❛❧s♦ ✐♥ ♣❛rt ♦❢ ❬✶✵❪✮✳ ❋✉rt❤❡r♠♦r❡✱ ❢♦r ❛t②♣✐❝❛❧ ✢♦✇ ✐♥ t❤❡ ♦♣❡♥ s❡t U¬min✱ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛r❡ ❛s②♠♠❡tr✐❝✱ s❡❡ ❙❡❝t✐♦♥ ✷✳✸✳

❖✉r ♠❛✐♥ r❡s✉❧t ✐♥ t❤❡ s❡t ✉♣ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❘❡❝❛❧❧ t❤❛t ❛♥ ■❊❚ T ✐s ❣✐✈❡♥ ❜② ❛❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t✉♠ π ❛♥❞ ❛ ❧❡♥❣❤t ❞❛t✉♠ λ ✇❤✐❝❤ ❞❡s❝r✐❜❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ♦r❞❡r ❛♥❞ ❧❡♥❣❤ts ♦❢ t❤❡❡①❝❤❛♥❣❡❞ s✉❜✐♥t❡r✈❛❧s ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✷✮✳ ❲❡ s❛② t❤❛t ❛ r❡s✉❧t ❤♦❧❞s ❢♦r ❛❧♠♦st ❡✈❡r② ■❊❚ ✐❢ ✐t ❤♦❧❞s❢♦r ❛♥② ✐rr❡❞✉❝✐❜❧❡ π ❛♥❞ ▲❡❜❡s❣✉❡ ❛❧♠♦st ❡✈❡r② ❝❤♦✐❝❡ ♦❢ λ ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✷✮✳

❚❤❡♦r❡♠ ✶✳✷✳ ❋♦r ❛❧♠♦st ❡✈❡r② ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥ T : I → I ❛♥❞ ❡✈❡r② r♦♦❢ ❢✉♥❝✲t✐♦♥ f : I → R

+ ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛t t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ T ✭✐♥ t❤❡ s❡♥s❡ ♦❢❉❡✜♥✐t✐♦♥ ✷✳✶✮✱ t❤❡ s♣❡❝✐❛❧ ✢♦✇ (ϕt)t∈R ♦✈❡r T ✉♥❞❡r f ✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✳

✶✳✸✳ P❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ❛♥❞ ❘❛t♥❡r ♣r♦♣❡rt✐❡s✳ ❚❤❡ r❡s✉❧t t❤❛t ✇❡ ✉s❡ ❛s ❛ ❝r✉❝✐❛❧ t♦♦❧ t♦♣r♦✈❡ ♠✉❧t✐♣❧❡ ♠✐①✐♥❣ ✐s t❤❛t t❤❡ ✢♦✇s t❤❛t ✇❡ ❝♦♥s✐❞❡r ✐♥ ❚❤❡♦r❡♠ ✶✳✶ ❛♥❞ ❚❤❡♦r❡♠ ✶✳✷ s❛t✐s❢② ❛✈❛r✐❛t✐♦♥ ♦❢ t❤❡ s♦ ❝❛❧❧❡❞ ❘❛t♥❡r ♣r♦♣❡rt② ♦❢ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡✳ ❲❡ ❜❡❧✐❡✈❡ t❤❛t t❤✐s ✐s ❛ r❡s✉❧t ♦❢✐♥❞❡♣❡♥❞❡♥t ✐♥t❡r❡st✱ s✐♥❝❡ ✐t ❞❡s❝r✐❜❡s ❛ ❝❡♥tr❛❧ ❢❡❛t✉r❡ ✇❤✐❝❤ s❤♦✇s t❤❡ ♣❛r❛❜♦❧✐❝ ❜❡❤❛✈✐♦✉r ♦❢ ✢♦✇s✇❡ st✉❞②✳ ❘❛t♥❡r ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✸✸❪ ❛ ♣r♦♣❡rt② t❤❛t s❤❡ ❝❛❧❧❡❞ Hp✲♣r♦♣❡rt② ❛♥❞ ✇❛s ❧❛t❡r ❦♥♦✇♥ ❛s❘❛t♥❡r ♣r♦♣❡rt② ✭❬✹✵❪✮✳ ❚❤✐s ♣r♦♣❡rt②✱ ✇❤♦s❡ ❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ✇❡ ♦♠✐t s✐♥❝❡ ✐t ✐s r❛t❤❡r t❡❝❤♥✐❝❛❧✭s❡❡ ❙❡❝t✐♦♥ ✷✳✹ ❢♦r ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥✮ ❞❡s❝r✐❜❡s ✐♥ ❛ ♣r❡❝✐s❡ q✉❛♥t❛t✐✈❡ ✇❛② ❤♦✇ ❢❛st ♥❡❛r❜②tr❛❥❡❝t♦r✐❡s ❞✐✈❡r❣❡✳ ❘❛t♥❡r ✜rst ✈❡r✐✜❡❞ t❤✐s ♣r♦♣❡rt② ❢♦r ❤♦r♦❝②❝❧❡ ✢♦✇s ❛♥❞ ✉s❡❞ ✐t t♦ ❞❡❞✉❝❡ s♦♠❡ ♦❢t❤❡ ♠❛✐♥ r✐❣✐❞✐t② ♣r♦♣❡rt✐❡s ♦❢ ❤♦r♦❝②❝❧❡ ✢♦✇s ✭s✉❝❤ ❛s ✈❡r② s♣❡❝✐✜❝ ♣r♦♣❡rt✐❡s ♦❢ ❥♦✐♥✐♥❣s ❛♥❞ ♠❡❛s✉r❡r✐❣✐❞✐t②✮✳ ❚❤❡ ❤♦r♦❝②❝❧❡ ✢♦✇ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ♠❛✐♥ ❡①❛♠♣❧❡ ✐♥ t❤❡ ❝❧❛ss ♦❢ ♣❛r❛❜♦❧✐❝ ✢♦✇s✱ ✐✳❡✳✐t ✐s ❛ ❝♦♥t✐♥✉♦✉s ❞②♥❛♠✐❝❛❧ s②st❡♠s ✐♥ ✇❤✐❝❤ ♥❡❛r❜② ♦r❜✐ts ❞✐✈❡r❣❡ ♣♦❧②♥♦♠✐❛❧❧②✳ ❚❤❡ ❘❛t♥❡r ♣r♦♣❡rt②❛s ♦r✐❣✐♥❛❧❧② ❞❡✜♥❡❞ ❜② ❘❛t♥❡r ❤♦❧❞s ❜② ✈✐rt✉❡ ♦❢ t❤✐s ♣♦❧②♥♦♠✐❛❧ ❞✐✈❡r❣❡♥❝❡✳

■t ✐s r❡❛s♦♥❛❜❧❡ t♦ ❡①♣❡❝t t❤❛t s♦♠❡ q✉❛♥t✐t❛t✐✈❡ ❢♦r♠ ♦❢ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ s✐♠✐❧❛r t♦ t❤❡ ❘❛t♥❡r♣r♦♣❡rt② s❤♦✉❧❞ ❤♦❧❞ ❛♥❞ ❜❡ ❝r✉❝✐❛❧ ✐♥ ♣r♦✈✐♥❣ ❛♥❛❧♦❣♦✉s r✐❣✐❞✐t② ♣r♦♣❡rt✐❡s ❢♦r ♦t❤❡r ❝❧❛ss❡s ♦❢ ♣❛r❛❜♦❧✐❝✢♦✇s✳ ❚❤✉s✱ t❤❡ ♥❛t✉r❛❧ q✉❡st✐♦♥ ❛r♦s❡✱ ✇❤❡t❤❡r t❤❡r❡ ❛r❡ ♣❛r❛❜♦❧✐❝ ✢♦✇s s❛t✐s❢②✐♥❣ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt②❜❡②♦♥❞ t❤❡ ❝❧❛ss ♦❢ ❤♦r♦❝②❝❧❡ ✢♦✇s✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ t❤✐s q✉❡st✐♦♥ ✇❛s ❣✐✈❡♥ ❜② ❑✳ ❋r→❝③❡❦ ❛♥❞▼✳ ▲❡♠❛➠❝③②❦ ✐♥ ❬✶✶❪✳ ❚❤❡ ❛✉t❤♦rs s❤♦✇❡❞ t❤❛t ❛ ✈❛r✐❛♥t ♦❢ ❘❛t♥❡r✬s ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞ ✐♥ t❤❡ ❝❧❛ss♦❢ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s ♦❢ ❝♦♥st❛♥t t②♣❡ ❛♥❞ ✉♥❞❡r s♦♠❡ r♦♦❢ ❢✉♥❝t✐♦♥s ♦❢ ❜♦✉♥❞❡❞

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✺

✈❛r✐❛t✐♦♥ ✭f(x) = ax + b✱ a, b > 0 ❜❡✐♥❣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡✮✳ ❚❤❡ ♥❡✇ ♣r♦♣❡rt②✱ ❝❛❧❧❡❞ t❤❡✜♥✐t❡ ❘❛t♥❡r✬s ♣r♦♣❡rt② ✐♥ ❬✶✶❪ ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✹ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥✮ ✇❛s s❤♦✇♥ t♦ ✐♠♣❧② t❤❡ s❛♠❡ ❥♦✐♥✐♥❣❝♦♥s❡q✉❡♥❝❡s ❛s t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡✳ ❚❤❡ ✜♥✐t❡ ❘❛t♥❡r ♣r♦♣❡rt② ✇❛s ❢✉rt❤❡r ✇❡❛❦❡♥❞ ❜② t❤❡ t✇♦ ❛✉t❤♦rs ✐♥❬✶✷❪ t♦ ✇❡❛❦ ❘❛t♥❡r✬s ♣r♦♣❡rt② ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✹✮✱ ✇❤✐❝❤ ✇❛s s❤♦✇♥ t✇♦ ❤♦❧❞ ✐♥ t❤❡ ❝❧❛ss ♦❢ s♣❡❝✐❛❧ ✢♦✇s♦✈❡r t✇♦✕❞✐♠❡♥s✐♦♥❛❧ r♦t❛t✐♦♥s ❛♥❞ s♦♠❡ r♦♦❢ ❢✉♥❝t✐♦♥s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ✭f(x, y) = ax + by + c✱a, b, c > 0 ❜❡✐♥❣ ♦♥❡ ❡①❛♠♣❧❡✮✳ ❆❧❧ t❤❡ ❞②♥❛♠✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❘❛t♥❡r✬s ♣r♦♣❡rt② ❤♦❧❞❛❧s♦ ❢♦r t❤❡ ✇❡❛❦ ❘❛t♥❡r ♣r♦♣❡rt②✱ ❬✶✷❪✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ r♦♦❢ ❜❡✐♥❣ ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ✇❛s❝r✉❝✐❛❧ ✐♥ ❬✶✶❪ ❛♥❞ ❬✶✷❪ ❛♥❞ ✉♥❢♦rt✉♥❛t❡❧② t❤✐s ❛ss✉♠♣t✐♦♥ ✐s ♥♦t ✈❡r✐✜❡❞ ❢♦r s♣❡❝✐❛❧ ✢♦✇ r❡♣r❡s❡♥t❛t✐♦♥s♦❢ ❆r♥♦❧❞ ✢♦✇s ❛♥❞ ♠♦r❡ ✐♥ ❣❡♥❡r❛❧ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✐♥ ❤✐❣❤❡r ❣❡♥✉s ✭s✐♥❝❡ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥❤❛s ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ❤❡♥❝❡ ✐s ♥♦t ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✮✳

■♥ ♣r❡s❡♥❝❡ ♦❢ s✐♥❣✉❧❛r✐t✐❡s s✉❝❤ ❛s t❤❡ ✜①❡❞ ♣♦✐♥ts ♦❢ s♠♦♦t❤ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇s✱ t❤❡ ❘❛t♥❡r♣r♦♣❡rt② ✐♥ ✐ts ❝❧❛ss✐❝❛❧ ❢♦r♠✱ ❛s ✇❡❧❧ ❛s t❤❡ ✇❡❛❦❡r ✈❡rs✐♦♥s ❞❡✜♥❡❞ ❜② ❑✳ ❋r→❝③❡❦ ❛♥❞ ▼✳ ▲❡♠❛➠❝③②❦✐s ❝✉rr❡♥t❧② ❡①♣❡❝t❡❞ t♦ ❢❛✐❧✳ ❚❤❡ ❤❡✉r✐st✐❝ ♣r♦❜❧❡♠ ❢♦r ❆r♥♦❧❞ ✢♦✇s ❛♥❞ ♠♦r❡ ❣❡♥❡r❛❧❧② s♠♦♦t❤ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇s t♦ ❡♥❥♦② t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ✭♦r ✐ts ✇❡❛❦❡r ✈❡rs✐♦♥s✮ ✐s t❤❛t ❘❛t♥❡r✲❧✐❦❡ ♣r♦♣❡rt✐❡sr❡q✉✐r❡ ❛ ✭♣♦❧②♥♦♠✐❛❧❧②✮ ❝♦♥tr♦❧❧❡❞ ✇❛② ♦❢ ❞✐✈❡r❣❡♥❝❡ ♦❢ ♦r❜✐ts ♦❢ ♥❡❛r❜② ♣♦✐♥ts✳ ■❢ t❤❡ ♦r❜✐ts ♦❢ t✇♦♥❡❛r❜② ♣♦✐♥ts ❣❡t t♦♦ ❝❧♦s❡ t♦ ❛ s✐♥❣✉❧❛r✐t②✱ t❤❡✐r ❞✐st❛♥❝❡ ❡①♣❧♦❞❡s ✐♥ ❛♥ ✉♥❝♦♥tr♦❧❧❡❞ ♠❛♥♥❡r✳ ❚❤✐s✐♥t✉✐t✐♦♥ ✇❛s s❤♦✇♥ t♦ ❜❡ ❝♦rr❡❝t ✐♥ ❬✶✵❪ ✭s❡❡ ❚❤❡♦r❡♠ ✶ ✐♥ t❤❡ ❆♣♣❡♥❞✐① ❇ ✐♥ ❬✶✵❪✮✱ ✇❤❡r❡ t❤❡ ✜rst❛✉t❤♦r s❤♦✇❡❞ t❤❛t s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ♦❢ ❝♦♥st❛♥t t②♣❡✱ ✉♥❞❡r ❛ r♦♦❢ ❢✉♥❝t✐♦♥ ♦❢ t❤❡❢♦r♠ f(x) = xγ ,−1 < γ < 0 ♦r f(x) = − log(x) ❞♦ ♥♦t s❛t✐s❢② t❤❡ ✇❡❛❦ ❘❛t♥❡r ♣r♦♣❡rt②✳

❚♦ ❞❡❛❧ ✇✐t❤ t❤✐s ✐ss✉❡s✱ ✐♥ ❬✶✵❪✱ ❇✳ ❋❛②❛❞ ❛♥❞ t❤❡ ✜rst ❛✉t❤♦r ✐♥tr♦❞✉❝❡❞ ❛ ♥❡✇ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡✇❡❛❦ ❘❛t♥❡r ♣r♦♣❡rt②✱ s♦ ❝❛❧❧❡❞ ❙❲❘✲♣r♦♣❡rt② ✭t❤❡ ❛❝r♦♥②♠ st❛♥❞✐♥❣ ❢♦r ❙✇✐t❝❤❛❜❧❡ ❲❡❛❦ ❘❛t♥❡r✮✱❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ ♦♥❡ ✐s ❛❧❧♦✇❡❞ t♦ ❝❤♦♦s❡ ✇❤❡t❤❡r ✇❡ s❡❡ ♣♦❧②♥♦♠✐❛❧ ❞✐✈❡r❣❡♥❝❡ ♦❢ ♦r❜✐ts ✐♥ t❤❡ ❢✉t✉r❡♦r ✐♥ t❤❡ ♣❛st✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ♣♦✐♥ts ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✷✳✷ ✐♥ ❙❡❝t✐♦♥ ✷✳✹✮✳ ❆❧❧ t❤❡ ❞②♥❛♠✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡s ♦❢t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ❛r❡ st✐❧❧ ✈❛❧✐❞ ❢♦r ❙❲❘✲♣r♦♣❡rt②✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ ♠✐①✐♥❣ ✢♦✇ ✇✐t❤ ❙❲❘✲♣r♦♣❡rt②✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✹✮✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ✐♥ ❬✶✵❪ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ✐s t❤❡❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✶✳✸✳ ❈♦♥s✐❞❡r t❤❡ s♣❡❝✐❛❧ ✢♦✇ (ϕt)t∈R ♦✈❡r ❛ r♦t❛t✐♦♥ Rα(x) = x + α mod 1 ❛♥❞ ✉♥❞❡r ❛r♦♦❢ ❢✉♥❝t✐♦♥ f : I → R

+ ✇✐t❤ ♦♥❡ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t② ❛t t❤❡ ③❡r♦✳ ❋♦r ❛❧♠♦st ❡✈❡r②α ∈ [0, 1]✱ (ϕt)t∈R s❛t✐s✜❡s t❤❡ ❙❲❘✲♣r♦♣❡rt② ❛♥❞ ❤❡♥❝❡ ✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ s❛♠❡r❡s✉❧t ❤♦❧❞s ✐❢ f ❤❛s s❡✈❡r❛❧ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✱ ✉♥❞❡r ❛ ♥♦♥ r❡s♦♥❛♥❝❡ ❝♦♥❞✐t✐♦♥ ✭♦❢❢✉❧❧ ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥✮ ❜❡t✇❡❡♥ t❤❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ t❤❡ ❜❛s❡ ❢r❡q✉❡♥❝② α✳

■♥ ♣❛rt✐❝✉❧❛r✱ s✐♥❝❡ ❢♦r ❆r♥♦❧❞ ✢♦✇s ♦♥ t♦r✐ ✭✐✳❡✳ t❤❡ r❡str✐❝t✐♦♥ ♦❢ ❛ s♠♦♦t❤ ❛r❡❛ ♣r❡s❡r✈✐♥❣ ✢♦✇ ♦♥❛ s✉r❢❛❝❡s ♦❢ ❣❡♥✉s ♦♥❡ t♦ ✐ts ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t✮✱ t❤❡ ❜❛s❡ ❛✉t♦♠♦r♣❤✐s♠ ✐♥ t❤❡ s♣❡❝✐❛❧ ✢♦✇ r❡♣r❡✲s❡♥t❛t✐♦♥ ♦❢ ❛♥ ❆r♥♦❧❞ ✢♦✇ ✐s ❛♥ ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ❛♥❞ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥ ❤❛s ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝s✐♥❣✉❧❛r✐t✐❡s✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❛❜♦✈❡ ❚❤❡♦r❡♠ t❤❛t t②♣✐❝❛❧❧② ❆r♥♦❧❞ ✢♦✇s ✇✐t❤ ♦♥❡ ✜①❡❞ ♣♦✐♥t s❛t✐s❢②t❤❡ ❙❲❘✲♣r♦♣❡rt② ❛♥❞ ❤❡♥❝❡ ❛r❡ t②♣✐❝❛❧❧② ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✳

■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ♣r♦✈❡ t❤❛t ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ❤♦❧❞s ❢♦r ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ♦❢t②♣✐❝❛❧ s♠♦♦t❤ ❛r❡❛ ♣r❡s❡r✈✐♥❣ ✢♦✇s ✐♥ U¬min ❢♦r s✉r❢❛❝❡s ♦❢ ❛♥② ❣❡♥✉s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ❝♦♥s✐❞❡r ❛str♦♥❣❡r ♣r♦♣❡rt②✱ t❤❡ ❙❘✲♣r♦♣❡rt② ✭❛❝r♦♥②♠ ❢♦r ❙✇✐t❝❤❛❜❧❡ ❘❛t♥❡r✱ ✇✐t❤♦✉t t❤❡ ❲ ❢♦r ❲❡❛❦ ✐♥ ❙❲❘✮✳❚❤✐s ♣r♦♣❡rt② tr✐✈✐❛❧❧② ✐♠♣❧✐❡s ❙❲❘✲♣r♦♣❡rt② ✭s❡❡ t❤❡ ❞❡✜♥✐t✐♦♥s ✐♥ ❙❡❝t✐♦♥ ✹✳✶✮✳ ❖✉r ♠❛✐♥ r❡s✉❧t ✐♥t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ✐s t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✶✳✹✳ ❋♦r ❛❧♠♦st ❡✈❡r② ■❊❚ T : I → I ❛♥❞ ❡✈❡r② r♦♦❢ ❢✉♥❝t✐♦♥ f : I → R+ ✇✐t❤ ❛s②♠♠❡tr✐❝

❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛t t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ T ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❉❡✜♥✐t✐♦♥ ✷✳✶✮✱ t❤❡ s♣❡❝✐❛❧ ✢♦✇(ϕt)t∈R ♦✈❡r T ❛♥❞ ✉♥❞❡r f ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✳

❆s ❛ ❈♦r♦❧❧❛r② ✭s❡❡ ❙❡❝t✐♦♥ ✺✳✸✮✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

❈♦r♦❧❧❛r② ✶✳✺✳ ❋♦r ❛♥② ✜①❡❞ ❣❡♥✉s g ≥ 1✱ ❝♦♥s✐❞❡r ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ♦♥ ❛ s✉r❢❛❝❡ S ♦❢ ❣❡♥✉sg ✇✐t❤ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts ❛♥❞ ❛t ❧❡❛st ♦♥❡ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥t✳ ❋♦r ❛ t②♣✐❝❛❧ ✢♦✇ (ϕt)t∈R ✐♥ ❛♥♦♣❡♥ ❛♥❞ ❞❡♥s❡ s❡t✱ t❤❡ r❡str✐❝t✐♦♥ ♦❢ (ϕt)t∈R t♦ ❛♥② ♦❢ ✐ts ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✳

✻ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

■t ✐s ❢r♦♠ t❤✐s r❡s✉❧t t❤❛t ✇❡ ❞❡❞✉❝❡ ❚❤❡♦r❡♠ ✶✳✷ ♦♥ ♠✉❧t✐♣❧❡ ♠✐①✐♥❣✱ s✐♥❝❡ t❤❡ ❙❲❘✲♣r♦♣❡rt② ✭❛♥❞❤❡♥❝❡ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ ❙❘✲♣r♦♣❡rt②✮ ❛❧❧♦✇s ✉s t♦ ❛✉t♦♠❛t✐❝❛❧❧② ✉♣❣r❛❞❡ ♠✐①✐♥❣ t♦ ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs❛♥❞ ♠✐①✐♥❣ ❢♦r t❤❡s❡ ✢♦✇s ✐s ❦♥♦✇♥ ❜② ❬✹✶✱ ✸✺❪✳

❚❤❡♦r❡♠ ✶✳✹ ❛♥❞ ❈♦r♦❧❧❛r② ✶✳✺ ❤❛✈❡ ❛❧s♦ ✐♠♣❧✐❝❛t✐♦♥s ♦♥ ❥♦✐♥✐♥❣ r✐❣✐❞✐t② ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✢♦✇s✭s✐♥❝❡ ❘❛t♥❡r ♣r♦♣❡rt✐❡s r❡str✐❝t t❤❡ ❝❧❛ss ♦❢ s❡❧❢✲❥♦✐♥✐♥❣s ❢♦r t❤❡s❡ ✢♦✇s✱ s❡❡ ❙❡❝t✐♦♥ ✷✳✹✮✳ ▼♦st ❝r✉❝✐❛❧❧②✱✐t s❤♦✇s t❤❡ ♣♦✇❡r ❛♥❞ ❣❡♥❡r❛❧✐t② ♦❢ t❤❡ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ✐♥tr♦❞✉❝❡❞ ❜② ❋❛②❛❞ ❛♥❞t❤❡ s❡❝♦♥❞ ❛✉t❤♦r ✐♥ ❝❛♣t✉r✐♥❣ t❤❡ q✉❛♥t✐t❛t✐✈❡ ❞✐✈❡r❣❡♥❝❡ ❜❡❤❛✈✐♦✉r ❢♦r ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ♣❛r❛❜♦❧✐❝ ✢♦✇s✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s✳ ❆ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❚❤❡♦r❡♠ ✶✳✷ ❛❧s♦ ✐♠♣❧✐❡s ✭❛s s❤♦✇♥ ✐♥ ❙❡❝t✐♦♥ ✺✳✸✮ t❤❡ ❢♦❧❧♦✇✐♥❣♥♦t❛❜❧❡ str❡♥❣❤t♥❡✐♥❣ ♦❢ t❤❡ ♠❛✐♥ r❡s✉❧t ✐♥ ❬✶✵❪✱ ✇❤✐❝❤ ✇❛s st❛t❡❞ ❤❡r❡ ❛s ❚❤❡♦r❡♠ ✶✳✸✳

❈♦r♦❧❧❛r② ✶✳✻✳ ❈♦♥s✐❞❡r t❤❡ s♣❡❝✐❛❧ ✢♦✇ (ϕt)t∈R ♦✈❡r ❛ r♦t❛t✐♦♥ Rα(x) = x + α mod 1 ❛♥❞ ✉♥❞❡r ❛r♦♦❢ ❢✉♥❝t✐♦♥ f : I → R

+ ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛t 0 < x1 < · · · < xd < 1 ✭✐♥ t❤❡s❡♥s❡ ♦❢ ❉❡✜♥✐t✐♦♥ ✷✳✶✮✳ ❋♦r ❛❧♠♦st ❡✈❡r② α ∈ [0, 1] ❛♥❞ ❛❧♠♦st ❡✈❡r② ❝❤♦✐❝❡ ♦❢ (x1, . . . , xd) ✇✐t❤ r❡s♣❡❝tt♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ [0, 1]d✱ (ϕt)t∈R s❛t✐s✜❡s t❤❡ ❙❘✲♣r♦♣❡rt②✳ ❍❡♥❝❡✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐t ✐s ♠✐①✐♥❣♦❢ ❛❧❧ ♦r❞❡rs✳

❲❡ r❡♠❛r❦ t❤❛t t❤❡ ❛❜♦✈❡ ❈♦r♦❧❧❛r② ❣❡♥❡r❛❧✐③❡s ❚❤❡♦r❡♠ ✶✳✸ ✐♥ t✇♦ ❞✐r❡❝t✐♦♥s✿ ✜rst ♦❢ ❛❧❧✱ ✐t s❤♦✇st❤❛t t❤❡ ✢♦✇s ❝♦♥s✐❞❡r❡❞ ✐♥ ❚❤❡♦r❡♠ ✶✳✸ ❤❛✈❡ t❤❡ str♦♥❣❡r ❙❘✲♣r♦♣❡rt② ✐♥st❡❛❞ t❤❛♥ t❤❡ ❙❲❘✲♣r♦♣❡rt②✳❙❡❝♦♥❞❧②✱ ❛t ♠♦st ✐♠♣♦rt❛♥t❧②✱ ♦✉r r❡s✉❧t ❤♦❧❞s ❢♦r ❛❧♠♦st ❡✈❡r② ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s✱ ✇❤✐❧❡ ✐♥❈♦r♦❧❧❛r② ✶✳✻ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s (x1, . . . , xd) ✇❛s r❡str✐❝t❡❞ t♦ ❛ s✉❜s❡t ♦❢ ❢✉❧❧ ❍❛✉s❞♦r✛❞✐♠❡♥s✐♦♥ ❜✉t ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ③❡r♦✳ ◆♦t✐❝❡ t❤♦✉❣❤✱ t❤❛t t❤❡ ❢✉❧❧ ♠❡❛s✉r❡ s❡t ✐♥ ❈♦r♦❧❧❛r② ✶✳✻ ✐s ♥♦t❡①♣❧✐❝✐t✱ ✇❤✐❧❡ t❤❡ r❡s♦♥❛♥❝❡ ❝♦♥❞✐t✐♦♥ ✐♥ ❚❤❡♦r❡♠ ✶✳✸ ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✶✳✸✳ ❛♥❞ ❘❡♠❛r❦ ✶✳✹✳ ✐♥ ❬✶✵❪ ❢♦r❞❡t❛✐❧s✮ ♠❛② ❛❧❧♦✇ t♦ ❝♦♥str✉❝t ❡①♣❧✐❝✐t ❡①❛♠♣❧❡s✳

✶✳✹✳ ❖✉t❧✐♥❡ ❛♥❞ str✉❝t✉r❡ ♦❢ t❤❡ ♣❛♣❡r✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♣r❡s❡♥t ❛♥ ♦✉t❧✐♥❡ ❛♥❞ s♦♠❡ ❤❡✉r✐st✐❝✐❞❡❛s ✉s❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✹✱ ♥❛♠❡❧② ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ✭✐♥ t❤❡ ♣r❡❝✐s❡ ❢♦r♠ ♦❢ t❤❡ s✇✐t❝❤❛❜❧❡❘❛t♥❡r ♣r♦♣❡rt② ❙❘✮ ❢♦r s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s ✉♥❞❡r r♦♦❢ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❤❛✈❡ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝s✐♥❣✉❧❛r✐t✐❡s ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❉❡✜♥✐t✐♦♥ ✷✳✶✮✱ s✐♥❝❡ t❤✐s ✐s t❤❡ ❦❡② r❡s✉❧t ❢r♦♠ ✇❤✐❝❤ t❤❡ ♦t❤❡r r❡s✉❧ts♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ❛r❡ t❤❡♥ ❞❡❞✉❝❡❞ ✭s❡❡ ❙❡❝t✐♦♥ ✺✳✸✮✳ ❚✇♦ ♦❢ t❤❡ ♠❛✐♥ ✐♥❣r❡❞✐❡♥ts ✉s❡❞✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✹ ❛r❡ ❛ s✉✐t❛❜❧❡ ❢✉❧❧ ♠❡❛s✉r❡ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ■❊❚ ♦♥ t❤❡ ❜❛s❡❛♥❞ ♣r❡❝✐s❡ q✉❛♥t✐t❛t✐✈❡ ❡st✐♠❛t❡s ♦♥ ❇✐r❦♦✛ s✉♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥s✳ ❲❡ ✇✐❧❧ ✜rst❝♦♠♠❡♥t ♦♥ t❤❡s❡ t✇♦ ♣❛rts✳

❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ♦♥ ■❊❚s ❛r❡ ❣✐✈❡♥ t❤r♦✉❣❤ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❛❧❣♦r✐t❤♠✱ ✇❤✐❝❤ ❝❛♥ ❜❡ t❤♦✉❣❤t♦❢ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r r♦t❛t✐♦♥s ✭s✐♥❝❡ r♦t❛t✐♦♥s ❝❛♥ ❜❡ s❡❡♥ ❛s ■❊❚s♦❢ t✇♦ ✐♥t❡r✈❛❧s✮✳ ❚❤✐s ❛ ♣♦✇❡r❢✉❧ ❛♥❞ ✇❡❧❧ st✉❞✐❡❞ t♦♦❧ t♦ ♣r♦✈❡ ✜♥❡ ♣r♦♣❡rt✐❡s ♦❢ ■❊❚s✱ ✇❤✐❝❤ ✇❛s❞❡✈❡❧♦♣❡❞ ❜② ❘❛✉③② ❬✸✹❪ ❛♥❞ ❱❡❡❝❤ ❬✹✺❪ ❛♥❞ ❤❛s ❜❡❡♥ ✉s❡❞ ✈❡r② ❢r✉✐t❢✉❧❧② ✐♥ t❤❡ ♣❛st t❤✐rt② ②❡❛rs✱ ❢♦r❡①❛♠♣❧❡✱ ❥✉st t♦ ♠❡♥t✐♦♥ ❛ ❢❡✇ ❤✐❣❤❧✐❣❤ts✱ t♦ ♣r♦✈❡ t❤❡ ♠❛✐♥ r❡s✉❧ts ✐♥ ❬✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✷✻✱ ✷✼✱ ✹✸✱ ✹✾❪❛♥❞ ♠❛♥② ♠♦r❡✳ ❚❤❡ ❘❛✉③②✲❱❡❡❝❤ ❛❧❣♦r✐t❤♠ ❛ss♦❝✐❛t❡s t♦ ❛♥ ■❊❚ ♦❢ d ✐♥t❡r✈❛❧s ❛ s❡q✉❡♥❝❡ ♦❢ d × d✐♥t❡❣❡r ✈❛❧✉❡❞ ♠❛tr✐❝❡s Aℓ ✇❤✐❝❤ ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❡♥tr✐❡s ♦❢ ❛ ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥❛❧❣♦r✐t❤♠✳ ❆s ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ❢♦r r♦t❛t✐♦♥s ❛r❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❣r♦✇t❤ ♦❢ t❤❡ ❝♦♥t✐♥✉❡❞❢r❛❝t✐♦♥ ❡♥tr✐❡s ♦❢ t❤❡ r♦t❛t✐♦♥ ♥✉♠❜❡r✱ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ❢♦r ■❊❚s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s♦❢ t❤❡ ❣r♦✇t❤ ♦❢ t❤❡ ♥♦r♠ ♦❢ t❤❡ ♠❛tr✐❝❡s Aℓ✳ ■t ✐s ❢r✉✐t❢✉❧ t♦ ❝♦♥s✐❞❡r ❛❝❝❡❧❡r❛t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❛r❡ ♣♦s✐t✐✈❡ ✭✐✳❡✳ t❤❡ ♠❛tr✐❝❡s Aℓ ❤❛✈❡ str✐❝t❧② ♣♦s✐t✐✈❡ ❡♥tr✐❡s✮ ❛♥❞ ❜❛❧❛♥❝❡❞ ✭✐✳❡✳ t✐♠❡s✇❤❡♥ t❤❡ ❘♦❤❧✐♥ t♦✇❡rs ✐♥ t❤❡ ❛ss♦❝✐❛t❡❞ ❞②♥❛♠✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ■❊❚ ❛s s✉s♣❡♥s✐♦♥ ♦✈❡r❛♥ ✐♥❞✉❝❡❞ ■❊❚ ❤❛✈❡ ❛♣♣r♦①✐♠❛t❡❧② t❤❡ s❛♠❡ ❤❡✐❣❤ts ❛♥❞ ✇✐❞t❤s✮✳

❖♥❡ ♦❢ t❤❡ ♠❛✐♥ ♣♦✐♥ts ♦❢ t❤✐s ♣❛♣❡r ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♥❡✇ ❉✐♦♣❤❛♥t✐♥❡ ❈♦♥❞✐t✐♦♥ ❢♦r ■❊❚s✱ t❤❛t✇❡ ❝❛❧❧ t❤❡ ❘❛t♥❡r ❉✐♦♣❤❛♥t✐♥❡ ❈♦♥❞✐t✐♦♥ ✭♦r ❘❛t♥❡r ❉❈ ❢♦r s❤♦rt✮✳ ❚❤✐s ✐♠♣❧✐❡s ❜② ❞❡✜♥✐t✐♦♥ t❤❡▼✐①✐♥❣ ❉❈ ❛♥❞ ✐t ✇❛s ✐♥s♣✐r❡❞ ❜② t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ❈♦♥❞✐t✐♦♥ ❢♦r r♦t❛t✐♦♥s ✐♥tr♦❞✉❝❡❞ ❜② ❋❛②❛❞ ❛♥❞t❤❡ ✜rst ❛✉t❤♦r ✐♥ ❬✶✵❪ ✭s❡❡ ❘❡♠❛r❦ ✸✳✺✮✳ ❚❤❡ ♣r♦♦❢ t❤❛t t❤❡ ❘❛t♥❡r ❉❈ ✐s s❛t✐s✜❡❞ ❢♦r ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t♦❢ ■❊❚s ❡①♣❧♦✐ts s✉❜t❧❡ ♣r♦♣❡rt✐❡s ♦❢ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥ ❛♥❞ ✐ts ♣♦s✐t✐✈❡ ❜❛❧❛♥❝❡❞ ❛❝❝❡❧❡r❛t✐♦♥s✱ ✐♥♣❛rt✐❝✉❧❛r ❛ q✉❛s✐✲❇❡r♥♦✉❧❧✐ t②♣❡ ♦❢ ♣r♦♣❡rt② ❛♥❞ t❤❡ ❢✉❧❧ str❡♥❣❤t ♦❢ t❤❡ ❞❡❡♣ ❡①♣♦♥❡♥t✐❛❧ t❛✐❧s ❡st✐♠❛t❡s❣✐✈❡♥ ❜② ❆✈✐❧❛✱ ●♦✉❡③❡❧ ❛♥❞ ❨♦❝❝♦③ ✐♥ ❬✸❪✳

▲❡t ✉s ♥♦✇ ❣✐✈❡ ❛♥ ✐♥t✉✐t✐✈❡ ❡①♣❧❛♥❛t✐♦♥ ♦❢ ✇❤② ❇✐r❦❤♦✛ s✉♠s ♦❢ ❞❡r✐✈❛t✐✈❡s ♣❧❛② ❛♥ ✐♠♣♦rt❛♥tr♦❧❡ ✐♥ ❜♦t❤ t❤❡ ♣r♦♦❢ ♦❢ ♠✐①✐♥❣ ❛♥❞ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ❢♦r s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s ✉♥❞❡r r♦♦❢s ✇✐t❤

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✼

❧♦❣❛r✐t❤♠✐❝ ❛s②♠♠❡tr✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ❙✐♥❝❡ T ✐s ❛ ♣✐❡❝❡✇✐s❡ ✐s♦♠❡tr② ✭❤❡♥❝❡ T ′ = 1 ❛❧♠♦st ❡✈❡r②✇❤❡r❡✮✱❜② t❤❡ ❝❤❛✐♥ r✉❧❡ ✇❡ ❤❛✈❡ t❤❛t

d

dxSr(f)(x) = Sr(f

′)(x) ❢♦r ❛✳❡✳ x, ❢♦r ❡❛❝❤ r ∈ Z.

❈♦♥s✐❞❡r ❛ s♠❛❧❧ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t J = [a, b] ✇❤✐❝❤ ✉♥❞❡r❣♦❡s ❡①❛❝t❧② r ❥✉♠♣s ✇❤❡♥ ✢♦✇✐♥❣ ❢♦r t✐♠❡t ✉♥❞❡r t❤❡ r♦♦❢ f ❛♥❞ ✇❤✐❝❤ ✐s ❛ ❝♦♥t✐♥✉✐t② ✐♥t❡r✈❛❧ ❢♦r T r✳ ❇② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❡①♣❧✐❝✐t ❡①♣r❡ss✐♦♥❢♦r t❤❡ s♣❡❝✐❛❧ ✢♦✇ ✐t❡r❛t❡s ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✷✮✱ ♦♥❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ s❡❣♠❡♥t J ❛❢t❡r t✐♠❡t ✐s ❣✐✈❡♥ ❜② (T r(x), Sr(f)(x)) ❢♦r x ∈ J ✳ ❚❤✉s✱ t❤❡ ❇✐r❦❤♦✛ s✉♠s Sr(f

′)(x) ❢♦r x ∈ J ❞❡s❝r✐❜❡ t❤❡✈❡rt✐❝❛❧ s❧♦♣❡ ♦❢ t❤❡ ✐♠❛❣❡ ♦❢ J ❛❢t❡r t✐♠❡ t ✉♥❞❡r t❤❡ ✢♦✇✳ ❚❤✐s s❧♦♣❡ ❝♦♥t❛✐♥s ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡s❤❡❛r✐♥❣ ♣❤❡♥♦♠❡♥♦♥ ✇❤✐❝❤ ✐s ❝r✉❝✐❛❧ ❜♦t❤ t♦ ♠✐①✐♥❣ ❛♥❞ t♦ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡✳ ❋♦r ❛♥ ❤❡✉r✐st✐❝❡①♣❧❛♥❛t✐♦♥ ♦❢ ❤♦✇ ♠✐①✐♥❣ ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ❜② s❤❡❛r✐♥❣✱ ✇❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ t❤❡ ♦✉t❧✐♥❡ ♦❢ ❬✹✶❪ ♦r❬✸✺❪❀ ❛ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❙❘✲Pr♦♣❡rt② ✉s✐♥❣ ❡st✐♠❛t❡s ♦♥ ❇✐r❦❤♦✛ s✉♠s ♦❢ ❞❡r✐✈❛t✐✈❡s ✐s ♣r❡s❡♥t❡❞ ✐♥❙❡❝t✐♦♥ ✹✳✶ ❛♥❞ ✇❛s ❛❧r❡❛❞② ✉s❡❞ ✐♥ ❬✶✺❪ ❛♥❞ ✐♥ ❛ s♣❡❝✐❛❧ ❝❛s❡ ✐♥ ❬✶✵❪✳

◆♦t❡ t❤❛t ✐❢ f ❤❛s ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✱ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ✐s ♥♦t ✐♥ L1(dx)✱ s✐♥❝❡ ✐t ❤❛s s✐♥❣✉❧❛r✐t✐❡s♦❢ t②♣❡ 1/x✱ ✇❤✐❝❤ ❛r❡ ♥♦t ✐♥t❡❣r❛❜❧❡✳ ❚❤✉s✱ ♦♥❡ ❝❛♥♥♦t ❛♣♣❧② t❤❡ ❇✐r❦❤♦✛ ❡r❣♦❞✐❝ t❤❡♦r❡♠ ✭✇❤✐❝❤✱ ❢♦r❛ ❢✉♥❝t✐♦♥ g ∈ L1(dx) ❣✉❛r❛♥t❡❡s t❤❛t Sr(g)/r ❝♦♥✈❡r❣❡ ♣♦✐♥t✇✐s❡ ❛❧♠♦st ❡✈❡r②✇❤❡r❡ t♦ ❛ ❝♦♥st❛♥t ❛♥❞t❤✉s t❤❛t Sr(g) ❣r♦✇s ❛s r✮✳ ❖♥❡ ❝❛♥ ✐♥❞❡❡❞ ♣r♦✈❡ t❤❛t✱ ❢♦r ❛ t②♣✐❝❛❧ ■❊❚ t❤❡ ❣r♦✇t❤ ♦❢ Sr(f

′) ✇❤❡♥ f❤❛s ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ✐s ♦❢ ♦r❞❡r Cr log r ✇❤❡r❡ C ✐s ❛ ❝♦♥st❛♥t ✇❤✐❝❤ ❞❡s❝r✐❜❡s t❤❡❛s②♠♠❡tr② ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s✳ ❚❤✐s ❛❞❞✐t✐♦♥❛❧ log r ❢❛❝t♦r ✐s r❡s♣♦♥s✐❜❧❡ ❢♦r t❤❡ s❤❡❛r✐♥❣ ♣❤❡♥♦♠❡♥♦♥❛t t❤❡ ❜❛s❡ ♦❢ ♠✐①✐♥❣ ❛♥❞ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t♦ ❝♦♥tr♦❧ t❤❡ ❣r♦✇t❤ ♦❢ Sr(f

′) ♣r❡❝✐s❡❧②✱♦♥❡ ♥❡❡❞s t♦ t❤r♦✇ ❛✇❛② ❛ s❡t ♦❢ ✐♥✐t✐❛❧ ♣♦✐♥ts ✇❤✐❝❤ ❝❤❛♥❣❡s ✇✐t❤ r✿ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ r ✐s ❜❡t✇❡❡♥ qℓ❛♥❞ qℓ+1✱ ✇❤❡r❡ qℓ ❞❡♥♦t❡s t❤❡ ♠❛①✐♠❛❧ ❤❡✐❣❤ts ♦❢ t♦✇❡rs ❛t st❡♣ nℓ ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❛❝❝❡❧❡r❛t✐♦♥✱♦♥❡ ♥❡❡❞s t♦ r❡♠♦✈❡ ❛ s❡t Σℓ ⊂ [0, 1] ✇❤♦s❡ ♠❡❛s✉r❡ ❣♦❡s t♦ ③❡r♦ ❛s ℓ ❣r♦✇s ✭s❡❡ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ❢♦rt❤❡ ♣r❡❝✐s❡ st❛t❡♠❡♥t✮✳

❲❡ ✉s❡ t❤❡s❡ s❤❛r♣ ❡st✐♠❛t❡s ♦♥ ❇✐r❦❤♦✛ s✉♠s ♦❢ ❞❡r✐✈❛t✐✈❡s ✭✐♥ t❤❡ ❢♦r♠ ♦❢ ▲❡♠♠❛ ✹✳✻✮ t♦ ♣r♦✈❡ t❤❡❙❘✲♣r♦♣❡rt② ♦❢ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡✳ ❲❡ ❢❛❝❡ t❤❡ ♣r♦❜❧❡♠✱ t❤♦✉❣❤✱ t❤❛t ✇❤✐❧❡ ♠✐①✐♥❣ ✐s ❛♥ ❛s②♠♣t♦t✐❝♣r♦♣❡rt②✱ ❛♥❞ ❤❡♥❝❡ r❡q✉✐r❡s ♦♥❧② t❤❛t s❤❡❛r✐♥❣ ❝❛♥ ❜❡ ❝♦♥tr♦❧❧❡❞ ❢♦r ❛r❜✐tr❛r✐❧② ❧❛r❣❡ t✐♠❡s r ♦✉ts✐❞❡♦❢ ❛ s❡t ✇❤♦s❡ ♠❡❛s✉r❡ ❣♦❡s t♦ ③❡r♦ ✭s♦ ✐t ✐s ❡♥♦✉❣❤ t♦ ✉s❡ t❤❛t t❤❡ ♠❡❛s✉r❡ ♦❢ Σℓ ❣♦❡s t♦ ③❡r♦ ✇✐t❤ ℓ✮✱t♦ ❝♦♥tr♦❧ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ♦♥❡ ♥❡❡❞s t♦ ❤❛✈❡ s❤❡❛r✐♥❣ ❢♦r ❛❧❧ ❛r❜✐tr❛r✐❧② ❧❛r❣❡ t✐♠❡s ❢♦r ♠♦st ♣♦✐♥ts✭✐✳❡✳ ♦♥ ❛ s❡t ♦❢ ❛r❜✐tr❛r✐❧② ❧❛r❣❡ ♠❡❛s✉r❡✮✳ ■❢ t❤❡ s❡r✐❡s ♦❢ t❤❡ ♠❡❛s✉r❡s ♦❢ Σℓ ✇❡r❡ s✉♠♠❛❜❧❡✱ t❛✐❧s✇♦✉❧❞ ❤❛✈❡ ❛r❜✐tr❛r✐❧② s♠❛❧❧ ♠❡❛s✉r❡s ❛♥❞ t❤✉s ♦♥❡ ❝♦✉❧❞ t❤r♦✇ ❛✇❛② t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts Σℓ ❢♦r ℓ❧❛r❣❡✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ♦♥❡ ❝❛♥ ❝❤❡❝❦ t❤❛t t❤❡ ♠❡❛s✉r❡s ♦❢ Σℓ ❛r❡ ♥♦t s✉♠♠❛❜❧❡✳

❚❤✐s ✐s ✇❤❡r❡ t❤❡ ❘❛t♥❡r ❉❈ ❤❡❧♣s✱ s✐♥❝❡✱ ✐❢ KT ⊂ N ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s ℓ s✉❝❤ t❤❛t✜♥✐t❡ ❜❧♦❝❦s ♦❢ ❝♦❝②❝❧❡ ♠❛tr✐❝❡s st❛rt✐♥❣ ✇✐t❤ Aℓ ❛r❡ ♥♦t t♦♦ ❧❛r❣❡ ✭♥♦t ❧❛r❣❡r t❤❛♥ ❛ ♣♦✇❡r ♦❢ log qℓ✱ s❡❡

✭✹✳✷✻✮ ✐♥ t❤❡ ❘❛t♥❡r ❉❈ ❞❡✜♥✐t✐♦♥ ❢♦r ❞❡t❛✐❧s✮✱ t❤❡ ❘❛t♥❡r ❉❈ ❣✉❛r❛♥t❡❡s t❤❛t t✐♠❡s ✐♥ N \ KT ✇❤❡r❡

t❤✐s ❢❛✐❧s ❛r❡ r❛r❡✱ ❛♥❞ ❤❡♥❝❡ ✐t ❝❛♥ ❜❡ ✉s❡❞ t♦ s❤♦✇ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ♠❡❛s✉r❡s ♦❢ Σℓ ❢♦r ℓ /∈ KT ✐s✜♥✐t❡ ✭s❡❡ ❈♦r♦❧❧❛r② ✹✳✽✮✳ ❚❤✉s✱ t❤❡s❡ s❡ts ❝❛♥ ❜❡ t❤r♦✇♥ ❢♦r ❧❛r❣❡ ℓ✳ ❖♥❡ ✐s t❤❡♥ ❧❡❢t t♦ ❡st✐♠❛t❡ t❤❡

t✐♠❡s ℓ ∈ KT ✳ ❚❤✐s ✐s ✇❤❡r❡ ♦♥❡ ❡①♣❧♦✐ts t❤❡ ✈❡rs❛t✐❧✐t② ♦❢ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt②✱ ❛❝❝♦r❞✐♥❣ t♦✇❤✐❝❤✱ ✐❢ t❤❡ ❞❡s✐r❡❞ q✉❛♥t✐✜❝❛t✐♦♥ ♦❢ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ❞♦❡s ♥♦t ❤♦❧❞ ❢♦r ❢♦r✇❛r❞ ❇✐r❦❤♦✛ s✉♠s ✭s❡❡✭✐✮ ✐♥ ❉❡✜♥✐t✐♦♥ ✷✳✷✮✱ ♦♥❡ ❝❛♥ s✇✐t❝❤ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t✐♠❡✱ ✐✳❡✳ ♣r♦✈❡ q✉❛♥t✐❛t✐✈❡ ❞✐✈❡r❣❡♥❝❡ ❡st✐♠❛t❡s♦♥ ❜❛❝❦✇❛r❞ ❇✐r❦❤♦✛ s✉♠s ✭s❡❡ ✭✐✐✮ ✐♥ ❉❡✜♥✐t✐♦♥ ✷✳✷✮✳ ❯s✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ❜❛❧❛♥❝❡❞ t✐♠❡s ✐♥ ❘❛✉③②✐♥❞✉❝t✐♦♥✱ ✇❡ s❤♦✇ t❤❛t ✐❢ ❛♥ ♦r❜✐t ♦❢ ❛ ♣♦✐♥t ♦❢ ❧❡♥❣❤t qℓ ❣❡ts t♦♦ ❝❧♦s❡ t♦ ❛ s✐♥❣✉❧❛r✐t② ✐♥ t❤❡ ❢✉t✉r❡✭✇❤❡r❡ t♦♦ ❝❧♦s❡ ✐s ♦❢ ♦r❞❡r qℓ+L ❢♦r ❛ ✜①❡❞ L✮✱ t❤❡♥ ✐t ❞✐❞ ♥♦t ❝♦♠❡ t❤❛t ❝❧♦s❡ t♦ ❛ s✐♥❣✉❧❛r✐t② ✐♥

t❤❡ ♣❛st ✭t❤✐s ✐s ♣r♦✈❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✺✳✶✮✳ ❚❤✉s✱ ✉s✐♥❣ t❤❛t ✐❢ ℓ ∈ KT t❤❡ ♥♦r♠s ♦❢ t❤❡ ❝♦❝②❝❧❡♠❛tr✐❝❡s Aℓ · · ·Aℓ+L ✐s ♥♦t t♦♦ ❧❛r❣❡ ✭❛♥❞ t❤r♦✇✐♥❣ ❛✇❛② ❛❞❞✐t✐♦♥❛❧ s❡ts ♦❢ ❜❛❞ ♣♦✐♥ts ✇❤♦s❡ ♠❡❛s✉r❡s❛r❡ s✉♠♠❛❜❧❡✮✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t ✐❢ ❇✐r❦❤♦✛ s✉♠s ❛r❡ ♥♦t ❝♦♥tr♦❧❧❡❞ ✐♥ t❤❡ ❢✉t✉r❡✱ t❤❡② ❛r❡ ❝♦♥tr♦❧❧❡❞✐♥ t❤❡ ♣❛st ✭▲❡♠♠❛ ✹✳✻✮✳ ❚❤✉s✱ t❤❡ ❝♦♥tr♦❧ r❡q✉✐r❡❞ ❜② t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ❤♦❧❞s ❢♦r ❛❧❧t✐♠❡s✳

❙tr✉❝t✉r❡ ♦❢ t❤❡ ♣❛♣❡r✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ s❡❝t✐♦♥s ❛r❡ ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷ ✇❡ r❡✈✐❡✇s♦♠❡ ❜❛❝❦❣r♦✉♥❞ ♠❛t❡r✐❛❧✱ ✐♥ ♣❛rt✐❝✉❧❛r ✇❡ ❣✐✈❡ t❤❡ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✭s❡❡❙❡❝t✐♦♥ ✷✳✶✮ ❛♥❞ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✷✮ ❛♥❞ ❡①♣❧❛✐♥ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠❡r t♦

✽ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

t❤❡ ❧❛tt❡r ✭✐♥ ❙❡❝t✐♦♥ ✷✳✸✮✳ ■♥ ❙❡❝t✐♦♥ ✷✳✸ ✇❡ ❛❧s♦ ❣✐✈❡ t❤❡ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝s✐♥❣✉❧❛r✐t✐❡s ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✷✳✶✮✳ ❲❡ t❤❡♥ ❞❡✜♥❡ ❘❛t♥❡r ♣r♦♣❡rt✐❡s✱ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ ❙❘✲♣r♦♣❡rt② ✇❡✉s❡ ✭s❡❡ ❉❡✜♥✐t✐♦♥s ✷✳✷ ❛♥❞ ✷✳✷ ✐♥ ❙❡❝t✐♦♥ ✷✳✹✮✳ ❋✐♥❛❧❧②✱ ✐♥ ❙❡❝t✐♦♥ ✷✳✺ ✇❡ r❡❝❛❧❧ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡❘❛✉③②✲❱❡❡❝❤ ❛❧❣♦r✐t❤♠ ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ❝♦❝②❝❧❡s✳ ❲❡ t❤❡♥ ❞❡s❝r✐❜❡ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥t❤❛t ✇❡ ✉s❡ ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✻✮ ❛♥❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✼ ✇❡ r❡❝❛❧❧ t❤❡ ❡①♣♦♥❡♥t✐❛❧ t❛✐❧ ❡st✐♠❛t❡s ❣✐✈❡♥ ❜② ❬✸❪✳

■♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ ❞❡✜♥❡ t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ♦♥ ■❊❚s ✇❤✐❝❤ ✇❡ ✉s❡ ✐♥ t❤✐s ♣❛♣❡r✱ ✐♥ ♣❛rt✐❝✉❧❛r✇❡ ✜rst r❡❝❛❧❧ t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✉♥❞❡r ✇❤✐❝❤ ♠✐①✐♥❣ ✇❛s ♣r♦✈❡❞ ✐♥ ❬✹✶❪ ❛♥❞ ❬✸✺❪ ✭s❡❡ t❤❡❉❡✜♥✐t✐♦♥ ✸✳✶ ♦❢ ▼✐①✐♥❣ ❉❈ ✐♥ ❙❡❝t✐♦♥ ✸✳✶✮✱ t❤❡♥ ✇❡ ❞❡✜♥❡ t❤❡ ❘❛t♥❡r ❉❈ ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✸✳✷ ✐♥❙❡❝t✐♦♥ ✸✳✷✮ ✉♥❞❡r ✇❤✐❝❤ ✇❡ ♣r♦✈❡ ♠✉❧t✐♣❧❡ ♠✐①✐♥❣ ❛♥❞ t❤❡ ❙❘✲❘❛t♥❡r ♣r♦♣❡rt②✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢t❤✐s s❡❝t✐♦♥ ✐s t❤❛t✱ ❢♦r ❛ s✉✐t❛❜❧❡ ❝❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs✱ t❤❡ ❘❛t♥❡r ❉❈ ✐s s❛t✐s✜❡❞ ❜② ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t♦❢ ■❊❚s ✭s❡❡ Pr♦♣♦s✐t✐♦♥ ✸✳✻✱ ✇❤✐❝❤ ✐s ♣r♦✈❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳✹ ✉s✐♥❣ t❤❡ ❡①♣♦♥❡♥t✐❛❧ t❛✐❧ ❡st✐♠❛t❡s r❡❝❛❧❧❡❞✐♥ ❙❡❝t✐♦♥ ✷✳✼ ❛♥❞ t❤❡ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ ◗❇✲♣r♦♣❡rt② ♦❢ ❝♦♠♣❛❝t ❛❝❝❡❧❡r❛t✐♦♥s ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤❝♦❝②❝❧❡ ♣r♦✈❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳✸✮✳

❇✐r❦❤♦✛ s✉♠s ❛♥❞ t❤❡✐r ❣r♦✇t❤ ❛r❡ t❤❡ ♠❛✐♥ ❢♦❝✉s ♦❢ ❙❡❝t✐♦♥ ✹✳ ■♥ ❙❡❝t✐♦♥ ✹✳✶✱ ✇❡ ✜rst r❡❝❛❧❧ ❛❝r✐t❡r✐✉♠ ✭❢r♦♠ ❬✶✵❪ ❛♥❞ ❬✶✺❪✮✱ ✇❤✐❝❤ ❛❧❧♦✇s t♦ r❡❞✉❝❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❙❘✲♣r♦♣❡rt② ❢♦r s♦♠❡ s♣❡❝✐❛❧✢♦✇s t♦ t❤❡ q✉❛♥t❛t✐✈❡ st✉❞② ♦❢ ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥✳ ■♥ ❙❡❝t✐♦♥ ✹✳✷ ✇❡ ✜rst st❛t❡ t❤❡❡st✐♠❛t❡s ♦♥ ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♣r♦✈❡❞ ✐♥ ❬✹✶✱ ✸✺❪ ✉♥❞❡r t❤❡ ▼✐①✐♥❣ ❉❈ ❛♥❞ t❤❡♥ ❞❡❞✉❝❡❡st✐♠❛t❡s ✐♥ ❢♦r♠ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❝♦♥✈❡♥✐❡♥t ❢♦r ✉s t♦ ♣r♦✈❡ t❤❡ ❙❲✲❘❛t♥❡r ♣r♦♣❡rt② ✭s❡❡ ▲❡♠♠❛ ✹✳✻✮✳❋✐♥❛❧❧②✱ ✐♥ ❙❡❝t✐♦♥ ✹✳✸ ✇❡ ❡①♣❧♦✐t t❤❡ ❘❛t♥❡r ❉❈ ❢♦r s✉✐t❛❜❧❡ ♣❛r❛♠❡t❡rs t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡ts Σl ✇✐t❤

l /∈ KT ✭s❡❡ t❤❡ ❛❜♦✈❡ ♦✉t❧✐♥❡✮ ❤❛✈❡ s✉♠♠❛❜❧❡ ♠❡❛s✉r❡s ✭s❡❡ t❤❡ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✐♥ ❉❡✜♥✐t✐♦♥✹✳✷ ❛♥❞ ❈♦r♦❧❧❛r② ✹✳✽✮✳

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ❛♥❞ ♦❢ t❤❡ ♦t❤❡r r❡s✉❧ts ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ✐♥tr♦❞✉❝t✐♦♥❛r❡ ❛❧❧ ❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✺✳ ❋✐rst✱ ✐♥ ❙❡❝t✐♦♥ ✺✳✶✱ ✇❡ ♣r♦✈❡ Pr♦♣♦s✐t✐♦♥ ✺✳✶ ✇❤✐❝❤ ❛❧❧♦✇s t♦ ❝♦♥tr♦❧ t❤❡❞✐st❛♥❝❡ ♦❢ ♦r❜✐ts ♦❢ ♠♦st ♣♦✐♥ts ❢r♦♠ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ❡✐t❤❡r ✐♥ t❤❡ ♣❛st ♦r ✐♥ t❤❡ ❢✉t✉r❡✳ ❚❤✐s ▲❡♠♠❛✱t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❡st✐♠❛t❡s ♦♥ ❇✐r❦❤♦✛ s✉♠s✱ ✐s t❤❡ ❧❛st ✐♥❣r❡❞✐❡♥t ♥❡❡❞❡❞❢♦r t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✹ ✭✐✳❡✳ t❤❡ ❙❘✲♣r♦♣❡rt② ❢♦r s♣❡❝✐❛❧ ✢♦✇s✮✱ ✇❤✐❝❤ ✐s ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✷✳❚❤❡ ♣r♦♦❢✱ ✇❤✐❝❤ ✐s r❛t❤❡r t❡❝❤♥✐❝❛❧✱ ✐s ♣r❡❝❡❡❞❡❞ ❜② ❛♥ ♦✉t❧✐♥❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❙❡❝t✐♦♥ ✺✳✷✳ ❚❤❡♦t❤❡r r❡s✉❧ts ✐♥ t❤✐s ✐♥tr♦❞✉❝t✐♦♥ ❛r❡ t❤❡♥ ♣r♦✈❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✸✳

❚❤❡ ❆♣♣❡♥❞✐① ❝♦♥t❛✐♥s ✐♥ ❙❡❝t✐♦♥ ❆✳✶ t❤❡ ♣r♦♦❢ t❤❛t ❘❛t♥❡r ♣r♦♣❡rt✐❡s ❛r❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t❛♥❞ ✐♥ ❙❡❝t✐♦♥ ❆✳✷ ❢♦r ❝♦♥✈❡♥✐❡♥❝❡ ♦❢ t❤❡ r❡❛❞❡r✱ t❤❡ ♣r♦♦❢ ♦❢ ❛ ♣r♦♣❡rt② ♦❢ ❜❛❧❛♥❝❡ ❘❛✉③②✲❱❡❡❝❤❛❝❝❡❧❡r❛t✐♦♥ t✐♠❡s t❤❛t ✇❛s ♣r♦✈❡❞ ✐♥ ❬✶✹❪ ❛♥❞ ✉s❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✶✳

✷✳ ❇❛❝❦❣r♦✉♥❞ ♠❛t❡r✐❛❧

✷✳✶✳ ▲♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s✳ ▲❡t (S, ω) ❜❡ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✱ ✐✳❡✳ S ✐s ❛❝❧♦s❡❞ ❝♦♥♥❡❝t❡❞ ♦r✐❡♥t❛❜❧❡ s♠♦♦t❤ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s g ≥ 1 ❛♥❞ ω ❛ ✜①❡❞ s♠♦♦t❤ ❛r❡❛ ❢♦r♠✳ ❆♥②s♠♦♦t❤ ❛r❡❛ ♣r❡s❡r✈✐♥❣ ✢♦✇ ♦♥ S ✐s ❣✐✈❡♥ ❜② ❛ s♠♦♦t❤ ❝❧♦s❡❞ r❡❛❧✲✈❛❧✉❡❞ ❞✐✛❡r❡♥t✐❛❧ 1✲❢♦r♠ η ❛s❢♦❧❧♦✇s✳ ▲❡t X ❜❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ❞❡t❡r♠✐♥❡❞ ❜② η = iXω = ω(η, ·) ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ✢♦✇ (ϕt)t∈R ♦♥ S❛ss♦❝✐❛t❡❞ t♦ X✳ ❙✐♥❝❡ η ✐s ❝❧♦s❡❞✱ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ϕt✱ t ∈ R✱ ❛r❡ ❛r❡❛✲♣r❡s❡r✈✐♥❣✳ ❈♦♥✈❡rs❡❧②✱ ❡✈❡r②s♠♦♦t❤ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✐♥ t❤✐s ✇❛②✳ ❚❤❡ ✢♦✇ (ϕt)t∈R ✐s ❦♥♦✇♥ ❛s t❤❡ ♠✉❧t✐✲✈❛❧✉❡❞❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ❛ss♦❝✐❛t❡❞ t♦ η✳ ■♥❞❡❡❞✱ t❤❡ ✢♦✇ (ϕt)t∈R ✐s ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥✱ ✐✳❡✳ ❧♦❝❛❧❧② ♦♥❡ ❝❛♥✜♥❞ ❝♦♦r❞✐♥❛t❡s (x, y) ♦♥ S ✐♥ ✇❤✐❝❤ (ϕt)t∈R ✐s ❣✐✈❡♥ ❜② t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❡q✉❛t✐♦♥s x = ∂H/∂y✱y = −∂H/∂x ❢♦r s♦♠❡ s♠♦♦t❤ r❡❛❧✲✈❛❧✉❡❞ ❍❛♠✐❧t♦♥✐❛♥ ❢✉♥❝t✐♦♥ H✳ ❆ ❣❧♦❜❛❧ ❍❛♠✐❧t♦♥✐❛♥ H ❝❛♥♥♦t❜❡ ✐♥ ❣❡♥❡r❛❧ ❜❡ ❞❡✜♥❡❞ ✭s❡❡ ❬✸✶❪✱ ❙❡❝t✐♦♥ ✶✳✸✳✹✮✱ ❜✉t ♦♥❡ ❝❛♥ t❤✐♥❦ ♦❢ (ϕt)t∈R ❛s ❣❧♦❜❛❧❧② ❣✐✈❡♥ ❜② ❛♠✉❧t✐✲✈❛❧✉❡❞ ❍❛♠✐❧t♦♥✐❛♥ ❢✉♥❝t✐♦♥✳

❖♥❡ ❝❛♥ ❞❡✜♥❡ ❛ t♦♣♦❧♦❣② ♦♥ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ❜② ❝♦♥s✐❞❡r✐♥❣ ♣❡rt✉r❜❛t✐♦♥s ♦❢ ❝❧♦s❡❞ s♠♦♦t❤1✲❢♦r♠s ❜② s♠♦♦t❤ ❝❧♦s❡❞ 1✲❢♦r♠s✳ ❲❡ ❛ss✉♠❡ t❤❛t 1✲❢♦r♠ η ✐s ▼♦rs❡✱ ✐✳❡✳ ✐t ✐s ❧♦❝❛❧❧② t❤❡ ❞✐✛❡r❡♥t✐❛❧♦❢ ❛ ▼♦rs❡ ❢✉♥❝t✐♦♥✳ ❚❤✉s✱ ❛❧❧ ③❡r♦s ♦❢ η ❝♦rr❡s♣♦♥❞ t♦ ❡✐t❤❡r ❝❡♥t❡rs ♦r s✐♠♣❧❡ s❛❞❞❧❡s✳ ❚❤✐s ❝♦♥❞✐t✐♦♥✐s ❣❡♥❡r✐❝ ✭✐♥ t❤❡ ❇❛✐r❡ ❝❛t❤❡❣♦r② s❡♥s❡✮ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ♣❡rt✉r❜❛t✐♦♥s ♦❢ ❝❧♦s❡❞ s♠♦♦t❤ 1✲❢♦r♠s ❜②❝❧♦s❡❞ s♠♦♦t❤ 1✲❢♦r♠s✳ ❆ ♠❡❛s✉r❡✲t❤❡♦r❡t✐❝❛❧ ♥♦t✐♦♥ ♦❢ t②♣✐❝❛❧ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ❜② ✉s✐♥❣ t❤❡ ❑❛t♦❦❢✉♥❞❛♠❡♥t❛❧ ❝❧❛ss ✭✐♥tr♦❞✉❝❡❞ ❜② ❑❛t♦❦ ✐♥ ❬✶✻❪✱ s❡❡ ❛❧s♦ ❬✸✶❪✮✱ ✐✳❡✳ t❤❡ ❝♦❤♦♠♦❧♦❣② ❝❧❛ss ♦❢ t❤❡ ✶✲❢♦r♠η ✇❤✐❝❤ ❞❡✜♥❡s t❤❡ ✢♦✇✳ ▲❡t Σ ❜❡ t❤❡ s❡t ♦❢ ✜①❡❞ ♣♦✐♥ts ♦❢ η ❛♥❞ ❧❡t k ❜❡ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ Σ✳ ▲❡t

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✾

γ1, . . . , γn ❜❡ ❛ ❜❛s❡ ♦❢ t❤❡ r❡❧❛t✐✈❡ ❤♦♠♦❧♦❣② H1(S,Σ,R)✱ ✇❤❡r❡ n = 2g+ k− 1✳ ❚❤❡ ✐♠❛❣❡ ♦❢ η ❜② t❤❡♣❡r✐♦❞ ♠❛♣ Per ✐s Per(η) = (

∫γ1η, . . . ,

∫γnη) ∈ R

n✳ ❚❤❡ ♣✉❧❧✲❜❛❝❦ Per∗Leb ♦❢ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡

❝❧❛ss ❜② t❤❡ ♣❡r✐♦❞ ♠❛♣ ❣✐✈❡s t❤❡ ❞❡s✐r❡❞ ♠❡❛s✉r❡ ❝❧❛ss ♦♥ ❝❧♦s❡❞ 1✲❢♦r♠s✳ ❲❤❡♥ ✇❡ ✉s❡ t❤❡ ❡①♣r❡ss✐♦♥t②♣✐❝❛❧ ❜❡❧♦✇✱ ✇❡ ♠❡❛♥ ❢✉❧❧ ♠❡❛s✉r❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ♠❡❛s✉r❡ ❝❧❛ss✳

▲❡t ✉s r❡❝❛❧❧ t❤❛t ❛ s❛❞❞❧❡ ❝♦♥♥❡❝t✐♦♥ ✐s ❛ ✢♦✇ tr❛❥❡❝t♦r② ❢r♦♠ ❛ s❛❞❞❧❡ t♦ ❛ s❛❞❞❧❡ ❛♥❞ ❛ s❛❞❞❧❡❧♦♦♣ ✐s ❛ s❛❞❞❧❡ ❝♦♥♥❡❝t✐♦♥ ❢r♦♠ ❛ s❛❞❞❧❡ t♦ t❤❡ s❛♠❡ s❛❞❞❧❡ ✭s❡❡ ❋✐❣✉r❡ ✶✭❛✮✮✳ ▲❡t ✉s r❡♠❛r❦ t❤❛t ✐❢t❤❡ ✢♦✇ (ϕt)t∈R ❣✐✈❡♥ ❜② ❛ ❝❧♦s❡❞ 1✲❢♦r♠ η ❤❛s ❛ s❛❞❞❧❡ ❧♦♦♣ ❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦ ✭✐✳❡✳ t❤❡ s❛❞❞❧❡ ❧♦♦♣✐s ❛ s❡♣❛r❛t✐♥❣ ❝✉r✈❡ ♦♥ t❤❡ s✉r❢❛❝❡✮✱ t❤❡♥ t❤❡ s❛❞❞❧❡ ❧♦♦♣ ✐s ♣❡rs✐st❡♥t ✉♥❞❡r s♠❛❧❧ ♣❡rt✉❜❛t✐♦♥s ✭s❡❡❙❡❝t✐♦♥ ✷✳✶ ✐♥ ❬✺✵❪ ♦r ▲❡♠♠❛ ✷✳✹ ✐♥ ❬✸✺❪✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ s❡t ♦❢ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✇❤✐❝❤ ❤❛✈❡❛t ❧❡❛st ♦♥❡ s❛❞❞❧❡ ❧♦♦♣ ✐s ♦♣❡♥✳ ❚❤❡ ♦♣❡♥ s❡ts Umin ❛♥❞ U¬min ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ❛r❡❞❡✜♥❡❞ r❡s♣❡❝t✐✈❡❧② ❛s t❤❡ ♦♣❡♥ s❡t U¬min t❤❛t ❝♦♥t❛✐♥s ❛❧❧ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✇✐t❤ s❛❞❞❧❡ ❧♦♦♣s❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦ ❛♥❞ t❤❡ ✐♥t❡r✐♦r Umin ✭✇❤✐❝❤ ♦♥❡ ❝❛♥ s❤♦✇ t♦ ❜❡ ♥♦♥✲❡♠♣t②✮ ♦❢ t❤❡ ❝♦♠♣❧❡♠❡♥t✱✐✳❡✳ t❤❡ s❡t ♦❢ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✇✐t❤♦✉t s❛❞❞❧❡ ❧♦♦♣s ❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦✶ ✭s❡❡ ❬✸✺❪ ❢♦r ❞❡t❛✐❧s✮✳

▲❡t ✉s r❡❝❛❧❧ ❢r♦♠ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ t❤❡ t♦♣♦❧♦❣✐❝❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇ ✐♥t♦♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ❛♥❞ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥ts✳ ❯♥❧❡ss t❤❡ s✉r❢❛❝❡ ✐s ♦❢ ❣❡♥✉s ♦♥❡ ❛♥❞ ❝♦♥s✐sts ♦❢ ❛✉♥✐q✉❡ ❝♦♠♣♦♥❡♥t✱ ❡❛❝❤ ❝♦♠♣♦♥❡♥t ✐s ❜♦✉♥❞❡❞ ❜② s❛❞❞❧❡ ❝♦♥♥❡❝t✐♦♥s✳ P❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥ts ❛r❡ ❡❧❧✐♣t✐❝✐s❧❛♥❞s ❛r♦✉♥❞ ❛ ❝❡♥t❡r ✭s❡❡ ❋✐❣✉r❡ ✶✭❛✮✮ ♦r ❝②❧✐♥❞❡rs ✜❧❧❡❞ ❜② ♣❡r✐♦❞✐❝ ♦r❜✐ts ✭s❡❡ ❋✐❣✉r❡ ✶✭❞✮✮✳ ❲❡r❡♠❛r❦ t❤❛t ✐❢ t❤❡ ✢♦✇ ✐s ♠✐♥✐♠❛❧✱ ✜①❡❞ ♣♦✐♥ts ❝❛♥ ❜❡ ♦♥❧② s❛❞❞❧❡s✱ s✐♥❝❡ ✐❢ t❤❡r❡ ✐s ❛ ❝❡♥t❡r✱ ✐t❛✉t♦♠❛t✐❝❛❧❧② ♣r♦❞✉❝❡s ❛♥ ✐s❧❛♥❞ ✜❧❧❡❞ ❜② ♣❡r✐♦❞✐❝ ♦r❜✐ts ❛♥❞ ❤❡♥❝❡ ❛ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥t✳ ■♥ t❤❡ ♦♣❡♥s❡t Umin ✇✐t❤ ♥♦ s❛❞❞❧❡ ❧♦♦♣s ❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦✱ ❛ t②♣✐❝❛❧ ✢♦✇ ❤❛s ♥♦ s❛❞❞❧❡ ❝♦♥♥❡❝t✐♦♥s ❛♥❞ t❤✐s✐♠♣❧✐❡s ♠✐♥✐♠❛❧✐t② ❜② ❛ r❡s✉❧t ♦❢ ▼❛✐❡r ❬✷✾❪ ✭♦r✱ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉s♣❡♥s✐♦♥ ✢♦✇s ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡♥❡①t s❡❝t✐♦♥✱ ❜② t❤❡ r❡s✉❧t ♦❢ ❑❡❛♥❡ ❬✶✼❪ ♦♥ ■❊❚s✮✳ ■♥ t❤❡ ♦♣❡♥ s❡t U¬min✱ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥ts ❛r❡t②♣✐❝❛❧❧② ❜♦✉♥❞❡❞ ❜② s❛❞❞❧❡ ❧♦♦♣s✳ ❆❢t❡r r❡♠♦✈✐♥❣ ❛❧❧ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥ts✱ ♦♥❡ ✐s t②♣✐❝❛❧❧② ❧❡❢t ✇✐t❤❝♦♠♣♦♥❡♥ts ✇✐t❤♦✉t s❛❞❞❧❡ ❝♦♥♥❡❝t✐♦♥s ♦♥ ✇❤✐❝❤ t❤❡ ✢♦✇ ✐s ♠✐♥✐♠❛❧ ✭❢♦r ❡①❛♠♣❧❡✱ ✐♥ ❋✐❣✉r❡ ✷✱ ❛❢t❡rr❡♠♦✈✐♥❣ ❛ ❝②❧✐♥❞❡r ❛♥❞ t✇♦ ✐s❧❛♥❞✱ ♦♥❡ ✐s ❧❡❢t ✇✐t❤ t✇♦ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ♦♥❡ ♦❢ ❣❡♥✉s ♦♥❡ ❛♥❞ ♦♥❡♦❢ ❣❡♥✉s t✇♦✮✳

✷✳✷✳ ❙♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s✳ ❆s ✇❡ ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✱ s♠♦♦t❤ ✢♦✇s ♦♥ ❤✐❣❤❡r ❣❡♥✉ss✉r❢❛❝❡s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✳ ▲❡t ✉s ✜rst r❡❝❛❧❧t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ■❊❚s ❛♥❞ ♦❢ s♣❡❝✐❛❧ ✢♦✇s✳■♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✳ ▲❡t I = I(0) = [0, 1) ❜❡ t❤❡ ✉♥✐t ✐♥t❡r✈❛❧✳ ❆♥ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡tr❛♥s❢♦r♠❛t✐♦♥ ✭■❊❚✮ ♦❢ d s✉❜✐♥t❡r✈❛❧s T : I → I ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t✉♠✷ π = (πt, πb)✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ❛ ♣❛✐r (πt, πb) ♦❢ ❜✐❥❡❝t✐♦♥s ❢r♦♠ A t♦ {1, . . . , d}✱ ✇❤❡r❡ d ≥ 2 ❛♥❞ A ✐s ❛ ✜♥✐t❡ s❡t ✇✐t❤d ❡❧❡♠❡♥ts ✭t, b st❛② ❤❡r❡ ❢♦r t♦♣ ❛♥❞ ❜♦tt♦♠ ♣❡r♠✉t❛t✐♦♥s✮ ❛♥❞ ❛ ❧❡♥❣t❤ ✈❡❝t♦r λ ✇❤✐❝❤ ❜❡❧♦♥❣ t♦ t❤❡

s✐♠♣❧❡① ∆d ♦❢ ✈❡❝t♦rs λ ∈ RA+ s✉❝❤ t❤❛t

∑α∈A λα = 1✳ ■♥❢♦r♠❛❧❧②✱ t❤❡ ✐♥t❡r✈❛❧ I(0) ✐s ❞❡❝♦♠♣♦s❡❞ ✐♥t♦

d ❞✐s❥♦✐♥t ✐♥t❡r✈❛❧s Iα ♦❢ ❧❡♥❣❤ts ❣✐✈❡♥ ❜② λα ❢♦r α ∈ A✳ ❚❤❡ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥ T ❣✐✈❡♥❜② (λ, π) ✐s ❛ ♣✐❡❝❡✇✐s❡ ✐s♦♠❡tr② t❤❛t r❡❛rr❛♥❣❡s t❤❡ s✉❜✐♥t❡r✈❛❧s ♦❢ ❧❡♥❣t❤s ❣✐✈❡♥ ❜② λ ✐♥ t❤❡ ♦r❞❡r❞❡t❡r♠✐♥❡❞ ❜② π✱ s♦ t❤❛t t❤❡ ✐♥t❡r✈❛❧s ❜❡❢♦r❡ t❤❡ ❡①❝❤❛♥❣❡✱ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ ❛r❡ Iπ−1

t (1), . . . , Iπ−1t (d)✱

✇❤✐❧❡ t❤❡ ♦r❞❡r ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ❛❢t❡r t❤❡ ❡①❝❤❛♥❣❡ ✐s Iπ−1b

(1), . . . , Iπ−1b

(d)✳ ❋♦r♠❛❧❧②✱ T ✱ ❢♦r ✇❤✐❝❤ ✇❡

s❤❛❧❧ ♦❢t❡♥ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ T = (λ, π)✱ ✐s t❤❡ ♠❛♣ T : I(0) → I(0) ❣✐✈❡♥ ❜②

Tx = x−∑

πb(β)<πb(α)

λβ +∑

πt(β)<πt(α)

λβ ❢♦r x ∈ I(0)α = [lα, rα),

✇❤❡r❡ lα =∑

πt(β)<πt(α)λβ ❛♥❞ rα = lα + λα ❢♦r α ∈ A ✭t❤❡ s✉♠s ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ❛r❡ ❜② ❝♦♥✈❡♥t✐♦♥

③❡r♦ ✐❢ ♦✈❡r t❤❡ ❡♠♣t② s❡t✱ ❡✳❣✳ ❢♦r α s✉❝❤ t❤❛t πt(α) = 1✮✳❲❡ s❛② t❤❛t T ✐s ♠✐♥✐♠❛❧ ✐❢ t❤❡ ♦r❜✐t ♦❢ ❛❧❧ ♣♦✐♥ts ❛r❡ ❞❡♥s❡✳ ❲❡ s❛② t❤❛t π = (πt, πb) ✐s ✐rr❡❞✉❝✐❜❧❡

✐❢ {1, . . . , j} ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r πb ◦ π−1t ♦♥❧② ❢♦r j = d✳ ■rr❡❞✉❝✐❜✐❧✐t② ✐s ❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥ ❢♦r

✶◆♦t❡ t❤❛t s❛❞❞❧❡ ❧♦♦♣s ♥♦♥ ❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦ ✭❛♥❞ s❛❞❞❧❡ ❝♦♥♥❡❝t✐♦♥s✮ ✈❛♥✐s❤ ❛❢t❡r ❛r❜✐tr❛r✐❧② s♠❛❧❧ ♣❡rt✉r❜❛t✐♦♥s❛♥❞ ♥❡✐t❤❡r t❤❡ s❡t ♦❢ ✶✲❢♦r♠s ✇✐t❤ s❛❞❞❧❡ ❧♦♦♣s ♥♦♥ ❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦ ✭♦r s❛❞❞❧❡ ❝♦♥♥❡❝t✐♦♥s✮ ♥♦r ✐ts ❝♦♠♣❧❡♠❡♥t ✐s♦♣❡♥✳

✷❲❡ ❛r❡ ✉s✐♥❣ ❤❡r❡ t❤❡ ♥♦t❛t✐♦♥ ❢♦r ■❊❚s ✐♥tr♦❞✉❝❡❞ ❜② ▼❛r♠✐✲▼♦✉ss❛✲❨♦❝❝♦③ ✐♥ ❬✷✻❪ ❛♥❞ s✉❜s❡q✉❡♥t❡❧② ✉s❡❞ ❜② ♠♦str❡❝❡♥t r❡❢❡r❡♥❝❡s ❛♥❞ ❧❡❝t✉r❡ ♥♦t❡s✳

✶✵ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

♠✐♥✐♠❛❧✐t②✳ ❘❡❝❛❧❧ t❤❛t T s❛t✐s✜❡s t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥ ✐❢ t❤❡ ♦r❜✐ts ♦❢ ❛❧❧ ❞✐s❝♦♥t✐♥✉✐t✐❡s lα ❢♦r α s✉❝❤t❤❛t πt(α) 6= 1 ❛r❡ ✐♥✜♥✐t❡ ❛♥❞ ❞✐s❥♦✐♥t✳ ■❢ T s❛t✐s✜❡s t❤✐s ❝♦♥❞✐t✐♦♥✱ t❤❡♥ T ✐s ♠✐♥✐♠❛❧ ❬✶✼❪✳❙♣❡❝✐❛❧ ✢♦✇s✳ ▲❡t T : I → I ❜❡ ❛♥ ■❊❚✳✸ ▲❡t f ∈ L1(I, dx) ❜❡ ❛ str✐❝t❧② ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ✇✐t❤∫I f(x) dx = 1✳ ▲❡t Xf + {(x, y) ∈ R

2 : x ∈ I(0), 0 ≤ y < f(x)} ❜❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ❜❡❧♦✇ t❤❡ ❣r❛♣❤♦❢ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥ f ❛♥❞ µ ❜❡ t❤❡ r❡str✐❝t✐♦♥ t♦ Xf ♦❢ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ dx dy✳ ●✐✈❡♥ x ∈ I ❛♥❞r ∈ N

+ ✇❡ ❞❡♥♦t❡ ❜②

✭✷✳✶✮ Sr(f)(x) +

∑r−1i=0 f(T

i(x)) ✐❢ r > 0;

0 ✐❢ r = 0;

−∑−1

i=r f(Ti(x)) ✐❢ r < 0;

t❤❡ rth ♥♦♥✲r❡♥♦r♠❛❧✐③❡❞ ❇✐r❦❤♦✛ s✉♠ ♦❢ f ❛❧♦♥❣ t❤❡ tr❛❥❡❝t♦r② ♦❢ x ✉♥❞❡r T ✳ ▲❡t t > 0✳ ●✐✈❡♥ x ∈ I✱❞❡♥♦t❡ ❜② r(x, t) t❤❡ ✐♥t❡❣❡r ✉♥✐q✉❡❧② ❞❡✜♥❡❞ ❜② r(x, t) + max{r ∈ N : Sr(f)(x) < t}✳

❚❤❡ s♣❡❝✐❛❧ ✢♦✇ ❜✉✐❧t ♦✈❡r T ✉♥❞❡r t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥ ❢ ✐s ❛ ♦♥❡✲♣❛r❛♠❡t❡r ❣r♦✉♣ (ϕt)t∈R ♦❢ µ✲♠❡❛s✉r❡♣r❡s❡r✈✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ Xf ✇❤♦s❡ ❛❝t✐♦♥ ✐s ❣✐✈❡♥✱ ❢♦r t > 0✱ ❜②

✭✷✳✷✮ ϕt(x, 0) =(T r(x,t)(x), t− Sr(x,t)(f)(x)

).

❋♦r t < 0✱ t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ✢♦✇ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ✐♥✈❡rs❡ ♠❛♣ ❛♥❞ ϕ0 ✐s t❤❡ ✐❞❡♥t✐t②✳ ❚❤❡ ✐♥t❡❣❡r r(x, t)❣✐✈❡s t❤❡ ♥✉♠❜❡r ♦❢ ❞✐s❝r❡t❡ ✐t❡r❛t✐♦♥s ♦❢ t❤❡ ❜❛s❡ tr❛♥s❢♦r♠❛t✐♦♥ T ✇❤✐❝❤ t❤❡ ♣♦✐♥t (x, 0) ✉♥❞❡r❣♦❡s✇❤❡♥ ✢♦✇✐♥❣ ✉♣ t♦ t✐♠❡ t > 0✳

✷✳✸✳ ▲♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ❛s s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s✳ ▲♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ❝❛♥ ❜❡r❡♣r❡s❡♥t❡❞ ❛s s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s ✉♥❞❡r r♦♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ❲❡ r❡❝❛❧❧♥♦✇ s♦♠❡ ♦❢ t❤❡ ❞❡t❛✐❧s ♦❢ t❤✐s r❡❞✉❝t✐♦♥❀ ❢♦r ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ s❡❡ ❬✸✺❪✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳ ▲❡t T : I → I ❜❡ ❛♥ ■❊❚✳ ❲❡ s❛② t❤❛t f : I → R∪ {+∞} ❤❛s ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s❛♥❞ ✇❡ ✇r✐t❡ f ∈ LogSing(T ) ✐❢✿

✭❛✮ f ✐s ❞❡✜♥❡❞ ♦♥ ❛❧❧ ♦❢ I \ {lα : α ∈ A}❀✭❜✮ f ∈ C2(I \ {lα : α ∈ A})❀✭❝✮ f ✐s ❜♦✉♥❞❡❞ ❛✇❛② ❢r♦♠ ③❡r♦❀✭❞✮ t❤❡r❡ ❡①✐st C+

α , C−α ≥ 0✱ α ∈ A s✉❝❤ t❤❛t

limx→l+α

f ′′(x)

(x− lα)−2= C+

α , limx→r−α

f ′′(x)

(rα − x)−2= C−

α .

▲❡t C+ +∑

αC+α ❛♥❞ C− +

∑αC

−α ❀ ✐❢ C

+ 6= C−✱ ✇❡ s❛② t❤❛t f ❤❛s ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✲✐t✐❡s ❛♥❞ ✇❡ ✇r✐t❡ f ∈ AsymLogSing(T )✳

❲❡ r❡♠❛r❦ t❤❛t ✐t ❢♦❧❧♦✇s ❢r♦♠ ✭❞✮ t❤❛t t❤❡ ❧♦❝❛❧ ❜❡❤❛✈✐♦✉r ♦❢ f ❝❧♦s❡ t♦ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ✐s f =C+α | log(x − lα)| + o(1) ❢♦r x → l+α ❛♥❞ f = C−

α | log(rα − x)| + o(1) ❢♦r x → r−α ✱ ❤❡♥❝❡ ✇❡ s♣❡❛❦ ♦❢❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ❲❡ r❡♠❛r❦ t❤❛t ✇❡ ❛❧❧♦✇ t❤❡ ♣♦ss✐❜✐❧✐t② t❤❛t s♦♠❡ C+

α ♦r C−α ❛r❡ ③❡r♦✱ s♦ f

❝♦✉❧❞ ❤❛✈❡ ❛ ✜♥✐t❡ ♦♥❡✲s✐❞❡❞ ❧✐♠✐t ❛t s♦♠❡ lα ♦r rα✱ ❜✉t ✇❡ ❛ss✉♠❡ t❤❛t ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s✐s ✐♥❞❡❡❞ ❧♦❣❛r✐t❤♠✐❝✳

▲❡t S′ ❜❡ ❛ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✢♦✇ (ϕt)t∈R ❞❡t❡r♠✐♥❡❞ ❜② η✳ ❚❤❡♥ ✇❡ ❝❛♥ ✜♥❞ ❛ s❡❣♠❡♥t Itr❛♥s✈❡rs❡ t♦ t❤❡ ✢♦✇ ❝♦♥t❛✐♥✐♥❣ ♥♦ ❝r✐t✐❝❛❧ ♣♦✐♥t ❛♥❞ s✉✐t❛❜❧❡ ❝♦♦r❞✐♥❛t❡s✱ s✉❝❤ t❤❛t t❤❡ ✜rst r❡t✉r♥♠❛♣ T : I → I ♦❢ (ϕt)t∈R t♦ I ✐s ❛♥ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥ T = (λ, π) ❡①❝❤❛♥❣✐♥❣ d ✐♥t❡r✈❛❧s✱✇❤❡r❡ d ✐s t❤❡ ♥✉♠❜❡r ♦❢ s❛❞❞❧❡ ♣♦✐♥ts ♦❢ η r❡str✐❝t❡❞ t♦ t❤❡ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t✳ ❙✐♥❝❡ S′ ✐s ❛ ♠✐♥✐♠❛❧❝♦♠♣♦♥❡♥t✱ π ✐s ✐rr❡❞✉❝✐❜❧❡✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡♠❛r❦ ✐s ✉s❡❢✉❧ t♦ s❤♦✇ t❤❛t ✐❢ ❛ ♣r♦♣❡rt② ❤♦❧❞s ❢♦r ❛❧♠♦st ❡✈❡r② ■❊❚ ♦♥ d ✐♥t❡r✈❛❧s✱✐t ❤♦❧❞s ❢♦r t❤❡ ✢♦✇ ❣✐✈❡♥ ❜② ❛ t②♣✐❝❛❧ η ♦♥ S✳

✸❖♥❡ ❝❛♥ ❞❡✜♥❡ ✐♥ t❤❡ s❛♠❡ ✇❛② s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ❛♥② ♠❡❛s✉r❡ ♣r❡s❡r✈✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥ T ♦❢ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡(M,M , µ)✱ s❡❡ ❡✳❣✳ ❬✾❪✳

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✶✶

❘❡♠❛r❦ ✷✳✶✳ ❖♥❡ ❝❛♥ ❝❤♦♦s❡ t❤❡ tr❛♥s✈❡rs❡ s❡❣♠❡♥t I s♦ t❤❛t t❤❡ ❧❡♥❣❤t ♦❢ ❡❛❝❤ ✐♥t❡r✈❛❧ Iα ❡①❝❤❛♥❣❡❞❜② T ❛♣♣❡❛rs ❛s ♦♥❡ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ Per(η)✱ ✇❤❡r❡ ✇❡ r❡❝❛❧❧ t❤❛t Per ❞❡♥♦t❡s t❤❡ ♣❡r✐♦❞ ♠❛♣❞❡✜♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✶✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ S1, . . . , Sκ ❛r❡ ❞✐st✐♥❝t ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✢♦✇ ❞❡t❡r✲♠✐♥❡❞ ❜② η✱ ♦♥❡ ❝❛♥ ❝❤♦♦s❡ tr❛♥s✈❡rs❡ s❡❣♠❡♥ts I1, . . . , Iκ ♦♥ ❡❛❝❤ Si ❛♥❞ ❝♦♦r❞✐♥❛t❡s ✐♥ ✇❤✐❝❤ t❤❡ ✜rstr❡t✉r♥ ♠❛♣s Ti : Ii → Ii ♦❢ (ϕt)t∈R t♦ Ii ❛r❡ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡s s✉❝❤ t❤❛t t❤❡ ❧❡♥❣❤ts ♦❢ t❤❡ ✐♥t❡r✈❛❧s❡①❝❤❛♥❣❡❞ ❜② Ti✱ ❢♦r 1 ≤ i ≤ κ✱ ❛❧❧ ❛♣♣❡❛r ❛s ❞✐st✐♥❝t ❝♦♦r❞✐♥❛t❡s ♦❢ Θ(η)✳

❚❤❡ ✜rst r❡t✉r♥ t✐♠❡ ❢✉♥❝t✐♦♥ f ♦♥ I ✭✐✳❡✳ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ s♣❡❝✐❛❧ ✢♦✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢(ϕt)t∈R✮ ❤❛s ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦❢ f ❛r❡ ❛s②♠♠❡tr✐❝✐s ♦♣❡♥ ❛♥❞ ❞❡♥s❡✿ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ♦♥❡ ❝❛♥ s❤♦✇ ✭s❡❡ ❬✸✺❪✮ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ♦♣❡♥ ❛♥❞ ❞❡♥s❡ s✉❜s❡tU′¬min ⊂ U¬min s✉❝❤ t❤❛t ❛❧❧ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ✐♥ U′

¬min ⊂ U¬min ❝❛♥❜❡ r❡♣r❡s❡♥t❡❞ ❛s s♣❡❝✐❛❧ ✢♦✇s ✉♥❞❡r ❛ r♦♦❢ ✐♥ AsymLogSing(T )✳

✷✳✹✳ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ♦❢ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡✳ ❲❡ r❡❝❛❧❧ ♥♦✇ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s✇✐t❝❤❛❜❧❡❘❛t♥❡r ♣r♦♣❡rt②✳ ❲❡ st❛t❡ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥ ✇❤✐❝❤ ❛❧s♦ ✐♥❝❧✉❞❡s t❤❡ ✇❡❛❦ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r♣r♦♣❡rt② ❛♥❞ t❤❡♥ ❝♦♠♠❡♥t ♦♥ t❤❡ ❞✐✛❡r❡♥❝❡s ✇✐t❤ t❤❡ ♦r✐❣✐♥❛❧ ❘❛t♥❡r ♣r♦♣❡rt② ✭s❡❡ ❘❡♠❛r❦ ✷✳✸ ❜❡❧♦✇✮✳❚❤❡ ❞❡✜♥✐t✐♦♥ ✐s r❛t❤❡r t❡❝❤♥✐❝❛❧ ❛♥❞ ✐s ❢♦❧❧♦✇❡❞ ❜② ❛♥ ✐♥t✉✐t✐✈❡ ❡①♣❧❛♥❛t✐♦♥ ♦❢ ✐ts ❤❡✉r✐st✐❝ ♠❡❛♥✐♥❣✭s❡❡ ❘❡♠❛r❦ ✷✳✷✮✳

▲❡t (X, d) ❜❡ ❛ σ✲❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✱ B t❤❡ σ✲❛❧❣❡❜r❛ ♦❢ ❇♦r❡❧ s✉❜s❡ts ♦❢ X✱ µ ❛ ❇♦r❡❧ ♣r♦❜❛❜✐❧✐t②♠❡❛s✉r❡ ♦♥ (X, d)✳ ▲❡t (Tt)t∈R ❜❡ ❛♥ ❡r❣♦❞✐❝ ✢♦✇ ❛❝t✐♥❣ ♦♥ (X,B, µ)✳

❉❡✜♥✐t✐♦♥ ✷✳✷ ✭❙❲❘✲Pr♦♣❡rt②✱ s❡❡ ❬✶✵❪✮✳ ❋✐① ❛ ❝♦♠♣❛❝t s❡t P ⊂ R \ {0} ❛♥❞ t0 > 0✳ ❲❡ s❛② t❤❛t t❤❡✢♦✇ (Tt)t∈R ❤❛s sR(t0, P )✲♣r♦♣❡rt② ✐❢✿

❢♦r ❡✈❡r② ε > 0 ❛♥❞ N ∈ N t❤❡r❡ ❡①✐st κ = κ(ε)✱ δ = δ(ε,N) ❛♥❞ ❛ s❡t Z = Z(ε,N) ✇✐t❤ µ(Z) > 1− ε✱s✉❝❤ t❤❛t✿

❢♦r ❡✈❡r② x, y ∈ Z ✇✐t❤ d(x, y) < δ ❛♥❞ x ♥♦t ✐♥ t❤❡ ♦r❜✐t ♦❢ y✱ t❤❡r❡ ❡①✐st p = p(x, y) ∈ P ❛♥❞M =M(x, y), L = L(x, y) ≥ N s✉❝❤ t❤❛t L

M ≥ κ ❛♥❞ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿

✭✐✮ 1L |{n ∈ [M,M + L] : d(Tnt0(x), Tnt0+p(y)) < ε}| > 1− ε✱

✭✐✐✮ 1L

∣∣{n ∈ [M,M + L] : d(T(−n)t0(x), T(−n)t0+p(y)) < ε}∣∣ > 1− ε✳

❲❡ s❛② t❤❛t (Tt)t∈R ❤❛s t❤❡ s✇✐t❝❤❛❜❧❡ ✇❡❛❦ ❘❛t♥❡r ♣r♦♣❡rt②✱ ♦r✱ ❢♦r s❤♦rt✱ t❤❡ ❙❲❘✲♣r♦♣❡rt② ✭✇✐t❤ t❤❡s❡t P ✮ ✐❢ {t0 > 0 : (Tt)t∈R ❤❛s sR(t0, P )✲♣r♦♣❡rt②} ✐s ✉♥❝♦✉♥t❛❜❧❡✳

❉❡✜♥✐t✐♦♥ ✷✳✸ ✭❙❲❘✲Pr♦♣❡rt②✱ s❡❡ ❬✶✵❪✮✳ ❲❡ s❛② t❤❛t t❤❡ ✢♦✇ (Tt)t∈R ❤❛s s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt②✱♦r✱ ❢♦r s❤♦rt✱ t❤❡ ❙❘✲♣r♦♣❡rt②✱ ✐❢ (Tt)t∈R ❤❛s t❤❡ ❙❲❘✲♣r♦♣❡rt② ✇✐t❤ t❤❡ s❡t P = {1,−1}✳

❘❡♠❛r❦ ✷✳✷✳ ■♥t✉✐t✐✈❡❧②✱ t❤❡ ❙❘✲♣r♦♣❡rt② ✭♦r t❤❡ ❙❲❘✲♣r♦♣❡rt②✮ ♠❡❛♥ t❤❛t✱ ❢♦r ❛ ❧❛r❣❡ s❡t ♦❢ ❝❤♦✐❝❡s♦❢ ♥❡❛r❜② ✐♥✐t✐❛❧ ♣♦✐♥ts ✭✐✳❡✳ ♣❛✐rs ♦❢ ♣♦✐♥ts ✐♥ t❤❡ s❡t Z ✇❤✐❝❤ ❛r❡ δ ❝❧♦s❡✮✱ t❤❡ ♦r❜✐ts ♦❢ t❤❡ t✇♦ ♣♦✐♥ts❡✐t❤❡r ✐♥ t❤❡ ♣❛st✱ ♦r ✐♥ t❤❡ ❢✉t✉r❡ ✭❛❝❝♦r❞✐♥❣ t♦ ✇❤❡❛t❤❡r ✭✐✮ ♦r ✭✐✐✮ ❤♦❧❞✮✱ ❞✐✈❡r❣❡ ❛♥❞ t❤❡♥✱ ❛❢t❡r s♦♠❡❛r❜✐tr❛r✐❧② ❧❛r❣❡ t✐♠❡ ✭Mt0 ♦r −Mt0✮ r❡❛❧✐❣♥✱ s♦ t❤❛t Tnt0(x) ✐s ❝❧♦s❡ t♦ ❛ ❛ s❤✐❢t❡❞ ♣♦✐♥t Tnt0+p(y) ♦❢t❤❡ ♦r❜✐t ♦❢ y ✭✇❤❡r❡ p ∈ P ❞❡♥♦t❡s t❤❡ t❡♠♣♦r❛❧ s❤✐❢t✮✱ ❛♥❞ t❤❡ t✇♦ ♦r❜✐ts t❤❡♥ st❛② ❝❧♦s❡ ❢♦r ❛ ✜①❡❞♣r♦♣♦rt✐♦♥ κ ♦❢ t❤❡ t✐♠❡ M ✳ ❖♥❡ ❝❛♥ s❡❡ t❤❛t t❤✐s t②♣❡ ♦❢ ♣❤❡♥♦♠❡♥♦♥ ✐s ♣♦ss✐❜❧❡ ♦♥❧② ❢♦r ♣❛r❛❜♦❧✐❝s②st❡♠s✱ ✐♥ ✇❤✐❝❤ ♦r❜✐ts ♦❢ ♥❡❛r❜② ♣♦✐♥ts ❞✐✈❡r❣❡ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ♦r s✉❜♣♦❧②♥♦♠✐❛❧ s♣❡❡❞✳

❘❡♠❛r❦ ✷✳✸✳ ❚❤❡ ♦r✐❣✐♥❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ❞✐✛❡rs ❢r♦♠ ❉❡✜♥✐t✐♦♥ ✷✳✸ ♦♥❧② ✐♥ t❤❛t ❢♦r❛❧❧ x, y ∈ Z ✭✐✮ ❤❛s t♦ ❜❡ s❛t✐s✜❡❞✳ ❚❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❝❤♦♦s✐♥❣✱ ❢♦r ❛ ❣✐✈❡♥ ♣❛✐r ♦❢ ♣♦✐♥ts✱ ✇❤❡t❤❡r ✭✐✮ ♦r✭✐✐✮ ❤♦❧❞s✱ ✐s t❤❡ r❡❛s♦♥ ✇❤② t❤❡ ♣r♦♣❡rt② ✇❛s ❝❛❧❧❡❞ s✇✐t❝❤❛❜❧❡ ❜② ❇✳ ❋❛②❛❞ ❛♥❞ t❤❡ ✜rst ❛✉t❤♦r ✐♥ ❬✶✵❪✿♦♥❡ ❝❛♥ s✇✐t❝❤ ❜❡t✇❡❡♥ ❡✐t❤❡r ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢✉t✉r❡ tr❛❥❡❝t♦r✐❡s ♦❢ t❤❡ ♣♦✐♥ts ✭✐❢ ✭✐✮ ❤♦❧❞s✮✱ ♦r t❤❡ ♣❛st✭✐❢ ✭✐✐✮ ❤♦❧❞s✮✳

▲❡t ✉s ❛❧s♦ str❡ss t❤❛t ✐♥ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ✭s♦♠❡ t✐♠❡s ❛❧s♦ ❝❛❧❧❡❞ t✇♦✲♣♦✐♥t ❘❛t♥❡r ♣r♦♣❡rt②✮P = {1,−1}✳ ❚❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❣✐✈❡♥ ❜② ❑✳ ❋r→❝③❡❦ ❛♥❞ ▼✳ ▲❡♠❛➠❝③②❦ ✐♥ ❬✶✶❪ ❛♥❞ ❬✶✷❪ ♠❡♥t✐♦♥❡❞ ✐♥t❤❡ ✐♥tr♦❞✉❝t✐♦♥✱ ✐✳❡✳ t❤❡ ✜♥✐t❡ ❘❛t♥❡r ♣r♦♣❡rt② ❛♥❞ t❤❡ ✇❡❛❦ ❘❛t♥❡r ♣r♦♣❡rt②✱ ❛♠♦✉♥t❡❞ t♦ ❛❧❧♦✇✐♥❣ Pt♦ ❜❡ ❛♥② ✜♥✐t❡ s❡t ♦r r❡s♣❡❝t✐✈❡❧② ❛♥② ❝♦♠♣❛❝t s❡t P ⊂ R \ {0}✳ ❚❤✉s t❤❡ ✇❡❛❦ ❘❛t♥❡r ♣r♦♣❡rt② ❬✶✷❪✐s ❛♥❛❧♦❣♦✉s t♦ ❉❡✜♥✐t✐♦♥ ✷✳✷ ❜✉t ✇✐t❤ t❤❡ r❡str✐❝t✐♦♥ t❤❛t ❢♦r ❛❧❧ x, y ∈ Z ✭✐✮ ❤❛s t♦ ❜❡ s❛t✐s✜❡❞✳

❆❧❧ t❤❡ ✈❛r✐❛♥ts ♦❢ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ❛r❡ ❞❡✜♥❡❞ s♦ t❤❛t t❤❡ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❚❤❡♦r❡♠ ✷✳✹❛♥❞ ❘❡♠❛r❦ ✷✳✺ ❤♦❧❞✳

✶✷ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

❚❤❡♦r❡♠ ✷✳✹ ✭❬✶✵❪✮✳ ▲❡t (X, d) ❜❡ ❛ σ✲❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✱ B t❤❡ σ✲❛❧❣❡❜r❛ ♦❢ ❇♦r❡❧ s✉❜s❡ts ♦❢ X✱µ ❛ ❇♦r❡❧ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ (X, d)✳ ▲❡t (Tt)t∈R ❜❡ ❛ ✢♦✇ ❛❝t✐♥❣ ♦♥ (X,B, µ)✳ ■❢ (Tt)t∈R ✐s ♠✐①✐♥❣❛♥❞ ❤❛s t❤❡ ❙❲❘✲♣r♦♣❡rt②✱ t❤❡♥ ✐t ✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✳

❘❡♠❛r❦ ✷✳✺✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t ✐❢ (Tt)t∈R ❤❛s t❤❡ ❙❲❘✲♣r♦♣❡rt② ✭❛♥❞ ❤❡♥❝❡ ✐♥♣❛rt✐❝✉❧❛r ✐❢ ✐t ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✮✱ t❤❡♥ ✐t ❤❛s ❛ ♣r♦♣❡rt② t❤❡ ✜♥✐t❡ ❡①t❡♥s✐♦♥ ♦❢ ❥♦✐♥✐♥❣s ♣r♦♣❡rt②✭s❤♦rt❡♥❡❞ ❛s ❋❊❏ ♣r♦♣❡rt②✮✱ ❬✶✵❪✱ ✇❤✐❝❤ ✐s ❛ r✐❣✐❞✐t② ♣r♦♣❡rt② t❤❛t r❡str✐❝ts t❤❡ t②♣❡ ♦❢ s❡❧❢✲❥♦✐♥✐♥❣st❤❛t (Tt)t∈R ❝❛♥ ❤❛✈❡ ❬✸✼✱ ✶✶❪✳ ❲❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬✶✸✱ ✸✼❪ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❥♦✐♥✐♥❣s ❛♥❞ ❋❊❏✳❋✉rt❤❡♠♦r❡✱ ✐t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t ✐❢ (Tt)t∈R ✐s ♠✐①✐♥❣ ❛♥❞ ❤❛s t❤❡ ❋❊❏ ♣r♦♣❡rt②✱ t❤❡♥ ✐t ✐s ❛✉t♦♠❛t✐❝❛❧❧②♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs✱ ❬✸✼❪✳

❖♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ ❙❘✲♣r♦♣❡rt②✱ ❛s ✇❡❧❧ ❛s ♦t❤❡r ❘❛t♥❡r ♣r♦♣❡rt✐❡s ✭✇✐t❤ t❤❡ s❡t P ❜❡✐♥❣ ✜♥✐t❡✮✱❛r❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t✳ ❲❡ ✐♥❝❧✉❞❡ t❤❡ ♣r♦♦❢ ♦❢ t❤✐s ❢❛❝t ✐♥ t❤❡ ❆♣♣❡♥❞✐① ❆✳✶ ✭s❡❡ ▲❡♠♠❛ ❆✳✶✮✳▲❡t ✉s r❡♠❛r❦ t❤❛t ✐t ✐s ♥♦t ❦♥♦✇♥ ✇❤❡t❤❡r t❤❡ ✇❡❛❦ ❘❛t♥❡r ♣r♦♣❡rt② ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t✳

✷✳✺✳ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥✳ ❚❤❡ ❘❛✉③②✲❱❡❡❝❤ ❛❧❣♦r✐t❤♠ ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ❘❛✉③②✲❱❡❡❝❤ ❝♦❝②❝❧❡✇❡r❡ ♦r✐❣✐♥❛❧❧② ✐♥tr♦❞✉❝❡❞ ❛♥❞ ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ✇♦r❦s ❜② ❘❛✉③② ❛♥❞ ❱❡❡❝❤ ❬✸✹✱ ✹✹✱ ✹✺❪ ❛♥❞ ♣r♦✈❡❞s✐♥❝❡ t❤❡♥ t♦ ❜❡ ❛ ♣♦✇❡r❢✉❧ t♦♦❧ t♦ st✉❞② ■❊❚s✳ ■❢ T = (λ, π) s❛t✐s✜❡s t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥ r❡❝❛❧❧❡❞ ✐♥❙❡❝t✐♦♥ ✷✳✷✱ ✇❤✐❝❤ ❤♦❧❞s ❢♦r ❛✳❡✳ ■❊❚ ❜② ❬✶✼❪✱ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❛❧❣♦r✐t❤♠ ♣r♦❞✉❝❡s ❛ s❡q✉❡♥❝❡ ♦❢ ■❊❚s

✇❤✐❝❤ ❛r❡ ✐♥❞✉❝❡❞ ♠❛♣s ♦❢ T ♦♥t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ♥❡st❡❞ s✉❜✐♥t❡r✈❛❧s ❝♦♥t❛✐♥❡❞ ✐♥ I(0)✳ ❚❤❡ ✐♥t❡r✈❛❧s❛r❡ ❝❤♦s❡♥ s♦ t❤❛t t❤❡ ✐♥❞✉❝❡❞ ♠❛♣s ❛r❡ ❛❣❛✐♥ ■❊❚s ♦❢ t❤❡ s❛♠❡ ♥✉♠❜❡r d ♦❢ ❡①❝❤❛♥❣❡❞ ✐♥t❡r✈❛❧s✳ ❋♦rt❤❡ ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ ✇❡ r❡❢❡r ❡✳❣✳ t♦ t❤❡ r❡❝❡♥t ❧❡❝t✉r❡ ♥♦t❡s ❜② ❨♦❝❝♦③ ❬✹✼❪ ♦r ❱✐❛♥❛❬✹✻❪✳ ❲❡ r❡❝❛❧❧ ❤❡r❡ ♦♥❧② s♦♠❡ ❜❛s✐❝ ❞❡✜♥✐t✐♦♥s ❛♥❞ ♣r♦♣❡rt✐❡s ♥❡❡❞❡❞ ✐♥ t❤❡ r❡st ♦❢ t❤✐s ♣❛♣❡r✳

▲❡t ✉s ❞❡♥♦t❡ ❜② | · | t❤❡ ✈❡❝t♦r ♥♦r♠ |λ| =∑

α∈A λα✳ ■❢ I ′ ⊂ I(0) ✐s t❤❡ s✉❜✐♥t❡r✈❛❧ ❛ss♦❝✐❛t❡❞ t♦♦♥❡ st❡♣ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥❞ T ′ ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥❞✉❝❡❞ ■❊❚✱ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♠❛♣ R ❛ss♦❝✐❛t❡st♦ T t❤❡ ■❊❚ R(T ) ♦❜t❛✐♥❡❞ ❜② r❡♥♦r♠❛❧✐③✐♥❣ T ′ ❜② Leb (I ′) s♦ t❤❛t t❤❡ r❡♥♦r♠❛❧✐③❡❞ ■❊❚ ✐s ❛❣❛✐♥❞❡✜♥❡❞ ♦♥ ❛♥ ✉♥✐t ✐♥t❡r✈❛❧✳ ❚❤❡ ♥❛t✉r❛❧ ❞♦♠❛✐♥ ♦❢ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♠❛♣ R ✐s ❛ ❢✉❧❧ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡s✉❜s❡t ♦❢ t❤❡ s♣❛❝❡ X + ∆d × R(π)✱ ✇❤❡r❡ R(π) ✐s t❤❡ ❘❛✉③② ❝❧❛ss ♦❢ π ✭✐✳❡✳ t❤❡ s✉❜s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢❜✐❥❡❝t✐♦♥s π′ = (π′t, π

′b) ❢r♦♠ A t♦ {1, . . . , d} ✇❤✐❝❤ ❛♣♣❡❛r ❛s ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t❛ ♦❢ ❛♥ ■❊❚ T ′ = (λ′, π′)

✐♥ t❤❡ ♦r❜✐t ✉♥❞❡r R ♦❢ s♦♠❡ ■❊❚ (λ′, π) ✇✐t❤ ✐♥✐t✐❛❧ ♣❛✐r ♦❢ ❜✐❥❡❝t✐♦♥s π = (πt, πb)✮✳ ❲❡ ✇✐❧❧ ❞❡♥♦t❡ ❜②∆π + ∆d × {π} t❤❡ ❝♦♣② ♦❢ t❤❡ s✐♠♣❧❡① ✐♥❞❡①❡❞ ❜② π✳

❱❡❡❝❤ ♣r♦✈❡❞ ✐♥ ❬✹✺❪ t❤❛t R ❛❞♠✐ts ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ µ = µR ✭✇❡ ✇✐❧❧ ✉s✉❛❧❧② s✐♠♣❧② ✇r✐t❡ µ✉♥❧❡ss ✇❡ ✇❛♥t t♦ str❡ss ✐s t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r R ❛♥❞ ♥♦t ❛♥② ♦❢ ✐ts ❛❝❝❡❧❡r❛t✐♦♥s ❞❡✜♥❡❞ ❜❡❧♦✇✮✇❤✐❝❤ ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✱ ❜✉t ✐♥✜♥✐t❡✳ ❩♦r✐❝❤ s❤♦✇❡❞ ✐♥ ❬✹✽❪t❤❛t ♦♥❡ ❝❛♥ ✐♥❞✉❝❡ t❤❡ ♠❛♣ R ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛♥ ❛❝❝❡❧❡r❛t❡❞ ♠❛♣ Z✱ ✇❤✐❝❤ ✇❡ ❝❛❧❧ ❩♦r✐❝❤ ♠❛♣✱ t❤❛t❛❞♠✐ts ❛ ✜♥✐t❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ µZ✳ ❇♦t❤ t❤❡s❡ ♠❡❛s✉r❡s ❤❛✈❡ ❛♥ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ❞❡♥s✐t② ✇✐t❤r❡s♣❡❝t t♦ t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ R

d t♦ ❡❛❝❤ ❝♦♣② ∆π ♦❢ t❤❡ s✐♠♣❧❡① ∆d✱ ✇❤✐❝❤ ✇❡✇✐❧❧ ❞❡♥♦t❡ ❜② LebX ✳ ▲❡t ✉s ❛❧s♦ r❡❝❛❧❧ t❤❛t ❜♦t❤ R ❛♥❞ ✐ts ❛❝❝❡❧❡r❛t✐♦♥ Z ❛r❡ ❡r❣♦❞✐❝ ✇✐t❤ r❡s♣❡❝t t♦µ = µR ❛♥❞ µZ r❡s♣❡❝t✐✈❡❧② ❬✹✺❪✳❘❛✉③②✲❱❡❡❝❤ ✭❧❡♥❣t❤s✮ ❝♦❝②❝❧❡✳ ❲❡ ✇✐❧❧ ♥♦✇ r❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝♦❝②❝❧❡ ❛ss♦❝✐❛t❡❞ ❜② t❤❡❛❧❣♦r✐t❤♠ t♦ t❤❡ ♠❛♣ R✳ ❋♦r ❡❛❝❤ T = (λ, π) ❢♦r ✇❤✐❝❤ R(T ) ✐s ❞❡✜♥❡❞✱ ✇❡ ❞❡✜♥❡ t❤❡ ♠❛tr✐①B = B(T ) ∈ SL(d,Z) s✉❝❤ t❤❛t λ = B · λ′✱ ✇❤❡r❡ λ′ s❛t✐s✜❡s R(T ) = (λ′/|λ′|, π′)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ |λ′| ✐st❤❡ ❧❡♥❣t❤ Leb(I ′) ♦❢ t❤❡ ✐♥❞✉❝✐♥❣ ✐♥t❡r✈❛❧ I ′ ♦♥ ✇❤✐❝❤ R(T ) ✐s ❞❡✜♥❡❞✳ ❚❤❡ ♠❛♣ B−1 : X → SL(d,Z)✐s ❛ ❝♦❝②❝❧❡ ♦✈❡r R✱ ❦♥♦✇♥ ❛s t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❝♦❝②❝❧❡✱ t❤❛t ❞❡s❝r✐❜❡s ❤♦✇ t❤❡ ❧❡♥❣t❤s tr❛♥s❢♦r♠✳ ■❢T = (λ, π) s❛t✐s✜❡s t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥ s♦ t❤❛t ✐ts ❘❛✉③②✲❱❡❡❝❤ ♦r❜✐t (Rn(T ))n∈N ✐s ✐♥✜♥✐t❡✱ ✇❡ ❞❡♥♦t❡

❜② T (n) + Rn(T ) t❤❡ ■❊❚ ♦❜t❛✐♥❡❞ ❛t t❤❡ nth st❡♣ ♦❢ ❘❛✉③②✲❱❡❡❝❤ ❛❧❣♦r✐t❤♠ ❛♥❞ ❜② (I(n))n∈N t❤❡

s❡q✉❡♥❝❡ ♦❢ ♥❡st❡❞ s✉❜✐♥t❡r✈❛❧s s♦ t❤❛t T (n) ✐s t❤❡ ✜rst r❡t✉r♥ ♠❛♣ ♦❢ T t♦ t❤❡ ✐♥t❡r✈❛❧ I(n) ⊂ I(0)✳❇② ❝♦♥str✉❝t✐♦♥✱ T (n) ✐s ❛❣❛✐♥ ❛♥ ■❊❚ ♦❢ d ✐♥t❡r✈❛❧s❀ ❧❡t π(n) ∈ R(π) ❛♥❞ λ(n) ∈ ∆ ❜❡ t❤❡ s❡q✉❡♥❝❡

♦❢ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛♥❞ ❧❡♥❣t❤s ❞❛t❛ s✉❝❤ t❤❛t T (n) = (λ(n)/|λ(n)|, π(n))✱ ✇❤❡r❡ |λ(n)| = Leb(I(n))✳ ■❢ ✇❡

❞❡✜♥❡ Bn = Bn(T ) + B(Rn(T )) ❛♥❞ B(n) = B(n)(T ) + B0 · . . . ·Bn−1 ❛♥❞ ✐t❡r❛t✐♥❣ t❤❡ ❧❡♥❣t❤s r❡❧❛t✐♦♥✱✇❡ ❣❡t

✭✷✳✸✮ λ(n) =(B(n)

)−1λ, ✇❤❡r❡ R

n(T ) +

(λ(n)

|λ(n)|, π(n)

).

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✶✸

❋♦r ♠♦r❡ ❣❡♥❡r❛❧ ♣r♦❞✉❝ts ✇✐t❤ m < n✱ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ B(m,n) + Bm ·Bm+1 · . . . ·Bn−1✳ ❚❤❡ ❡♥tr✐❡s

♦❢ B(m,n) ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞②♥❛♠✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✿ B(m,n)αβ ✐s ❡q✉❛❧ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ✈✐s✐ts ♦❢ t❤❡

♦r❜✐t ♦❢ ❛♥② ♣♦✐♥t x ∈ I(n)β t♦ t❤❡ ✐♥t❡r✈❛❧ I

(m)α ✉♥❞❡r t❤❡ ♦r❜✐t ♦❢ T (m) ✉♣ t♦ ✐ts ✜rst r❡t✉r♥ t♦ I(n)✳ ■♥

♣❛rt✐❝✉❧❛r✱∑

α∈AB(n)αβ ❣✐✈❡s t❤❡ ✜rst r❡t✉r♥ t✐♠❡ ♦❢ x ∈ I

(n)β t♦ I(n) ✉♥❞❡r T ✳

❘♦❤❧✐♥ t♦✇❡rs ❛♥❞ ❤❡✐❣❤ts ❝♦❝②❝❧❡✳ ❚❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ■❊❚ T = T (0) ❝❛♥ ❜❡ s❡❡♥ ✐♥ t❡r♠s ♦❢ ❘♦❤❧✐♥

t♦✇❡rs ♦✈❡r T (n) = Rn(T ) ❛s ❢♦❧❧♦✇s✳ ▲❡t h(n) ∈ NA ❜❡ t❤❡ ✈❡❝t♦r s✉❝❤ t❤❛t h

(n)β ❣✐✈❡s t❤❡ r❡t✉r♥ t✐♠❡

♦❢ ❛♥② x ∈ I(n)β t♦ I(n)✱ β ∈ A✳ ❇② t❤❡ ❛❜♦✈❡ ❞②♥❛♠✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✱ h

(n)β =

∑α∈AB

(n)αβ ✐s t❤❡ ♥♦r♠

♦❢ t❤❡ βth ❝♦❧✉♠♥ ♦❢ B(n)✳ ❉❡✜♥❡ t❤❡ s❡ts

✭✷✳✹✮ Z(n)α +

h(n)α −1⋃

i=0

T iI(n)α , α ∈ A.

❊❛❝❤ Z(n)α ❝❛♥ ❜❡ ✈✐s✉❛❧✐③❡❞ ❛s ❛ t♦✇❡r ♦✈❡r I

(n)α ⊂ I(n)✱ ♦❢ ❤❡✐❣❤t h

(n)α ✱ ✇❤♦s❡ ✢♦♦rs ❛r❡ T iI

(n)α ✳ ❯♥❞❡r

t❤❡ ❛❝t✐♦♥ ♦❢ T ❡✈❡r② ✢♦♦r ❜✉t t❤❡ t♦♣ ♦♥❡✱ ✐✳❡✳ ✐❢ 0 ≤ i < h(n)α − 1✱ ♠♦✈❡s ♦♥❡ st❡♣ ✉♣✱ ✇❤✐❧❡ t❤❡ ✐♠❛❣❡

❜② T ♦❢ t❤❡ ❧❛st ✢♦♦r✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ i = h(n)α − 1✱ ✐s T (n)I

(n)α ✳

❚❤❡ ❤❡✐❣❤t ✈❡❝t♦r h(n) ✇❤✐❝❤ ❞❡s❝r✐❜❡s r❡t✉r♥ t✐♠❡ ❛♥❞ ❤❡✐❣❤ts ♦❢ t❤❡ ❘♦❤❧✐♥ t♦✇❡rs ❛t st❡♣ n ♦❢ t❤❡✐♥❞✉❝t✐♦♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❛♣♣❧②✐♥❣ t❤❡ ❞✉❛❧ ❝♦❝②❝❧❡ BT ✱ t❤❛t ✇❡ ✇✐❧❧ ❝❛❧❧ ❧❡♥❣❤ts ❝♦❝②❝❧❡✱ ✐✳❡✳✱ ✐❢ h(0)

✐s t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦r ✇✐t❤ ❛❧❧ ❡♥tr✐❡s ❡q✉❛❧ t♦ 1✱

✭✷✳✺✮ h(n) = (BT )(n)h(0).

▲❡t ✉s ❞❡♥♦t❡ ❜② φ(n) ❜❡ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ I(0) ✐♥t♦ ✢♦♦rs ♦❢ st❡♣ n✱ ✐✳❡✳ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ❢♦r♠ T iI(n)α ✳

❲❤❡♥ T s❛t✐s✜❡s t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥✱ t❤❡ ♣❛rt✐t✐♦♥s φ(n) ❝♦♥✈❡r❣❡ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t② t♦ t❤❡ tr✐✈✐❛❧♣❛rt✐t✐♦♥s ✐♥t♦ ♣♦✐♥ts ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❬✹✼✱ ✹✻❪✮✳

◆❛t✉r❛❧ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥✳ ❚❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ R ♦❢ t❤❡ ♠❛♣ R ✐s ❛♥ ✐♥✈❡rt✐❜❧❡♠❛♣ ❞❡✜♥❡❞ ♦♥ ❛ ❞♦♠❛✐♥ X ✭✇❤✐❝❤ ❛❞♠✐ts ❛ ❣❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ③✐♣♣❡r❡❞

r❡❝t❛♥❣❧❡s✱ s❡❡ ❢♦r ❡①❛♠♣❧❡ ❬✹✼✱ ✹✻❪✮ s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣r♦❥❡❝t✐♦♥ p : X → X ❢♦r ✇❤✐❝❤ pR = Rp✳▼♦r❡ ♣r❡❝✐s❡❧②✱ ❢♦r ❛♥② π ✐♥ t❤❡ ❘❛✉③② ❝❧❛ss✱ ❧❡t Θπ ⊂ R

d+ ❜❡ t❤❡ s❡t ♦❢ ✈❡❝t♦rs τ s✉❝❤ t❤❛t

∑

πt(α)<j

τj > 0 ❛♥❞∑

πb(α)<j

τj < 0 ❢♦r 1 ≤ j ≤ d− 1.

❱❡❝t♦rs ✐♥ Θπ ❛r❡ ❝❛❧❧❡❞ s✉s♣❡♥s✐♦♥ ❞❛t❛✳ P♦✐♥ts ✐♥ X ❛r❡ tr✐♣❧❡s T = (τ, λ, π) s✉❝❤ t❤❛t

✭✷✳✻✮∑

α

λαhα = 1, ✇❤❡r❡ hα +∑

πt(β)<α,πb(β)>α

τβ −∑

πt(β)>α,πb(β)<α

τβ .

❚♦ ❡❛❝❤ s✉❝❤ tr✐♣❧❡ T = (τ, π, λ) ♦♥❡ ❝❛♥ ❛ss♦❝✐❛t❡ ❛ ❣❡♦♠❡tr✐❝ ♦❜❥❡❝t ❦♥♦✇♥ ❛s ③✐♣♣❡r❡❞ r❡❝t❛♥❣❧❡✳❲❡ r❡❢❡r t♦ ❬✹✼✱ ✹✻❪ ❢♦r ❞❡t❛✐❧s✳ ❚❤❡ ✈❡❝t♦r h = (hα)α∈A ❣✐✈❡s t❤❡ ❤❡✐❣❤ts ♦❢ t❤❡ r❡❝t❛♥❣❧❡s ❛♥❞ t❤❡✈❡❝t♦r λ = (λα) ❣✐✈❡s t❤❡✐r ❧❡♥❣❤ts ✭✇❤✐❧❡ τ ❝♦♥t❛✐♥ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❤♦✇ t♦ ③✐♣ t❤❡ ✈❡rt✐❝❛❧ s✐❞❡s ♦❢t❤❡ r❡❝t❛♥❣❧❡s t♦❣❡t❤❡r✮✳ ❚❤✉s✱ t❤❡ ❛❜♦✈❡ ♥♦r♠❛❧✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ✭✷✳✻✮ ❣✉❛r❛♥t❡❡s t❤❛t t❤❡ ❛ss♦❝✐❛t❡❞③✐♣♣❡r❡❞ r❡❝t❛♥❣❧❡ ❤❛s ❛r❡❛ ♦♥❡✳

❚❤❡ ♣r♦❥❡❝t✐♦♥ p ✐s ❞❡✜♥❡❞ ❜② p(τ, π, λ) = (π, λ)✳ ❚❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ R ♣r❡s❡r✈❡s ❛ ♥❛t✉r❛❧✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ µ✱ ✇❤♦s❡ ♣✉s❤✲❢♦r✇❛r❞ p∗µ ❜② t❤❡ ♣r♦❥❡❝t✐♦♥ p ✭✐✳❡✳ t❤❡ ♠❡❛s✉r❡ s✉❝❤ t❤❛t p∗µ(E) =µ(p−1E) ❢♦r ❛♥② ♠❡❛s✉r❛❜❧❡ s❡t ♦♥ X✮ ❡q✉❛❧s µ✳

❇♦t❤ ❝♦❝②❝❧❡s B−1 ❛♥❞ BT ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❝♦❝②❝❧❡s ♦✈❡r (X, µR, R) ✭❢♦r ✇❤✐❝❤ ✇❡ ✇✐❧❧ ✉s❡ t❤❡

s❛♠❡ ♥♦t❛t✐♦♥ B−1✱ BT ✮ ❜② s❡tt✐♥❣ B(τ, λ, π) + B(λ, π) ❢♦r ❛♥② (τ, λ, π) ∈ X✱ ✐✳❡✳ t❤❡ ❡①t❡♥❞❡❞ ❝♦❝②❝❧❡s❛r❡ ❝♦♥st❛♥t ♦♥ t❤❡ ✜❜❡rs ♦❢ p✳

✶✹ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

❈②❧✐♥❞❡r s❡ts✳ ▲❡t ✉s ❞❡✜♥❡ s②♠❜♦❧✐❝ ❝②❧✐♥❞❡rs ❢♦r t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♠❛♣ R ❛♥❞ ❢♦r ✐ts ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥

R✳ ❲❡ ✇✐❧❧ s❛② t❤❛t ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ♠❛tr✐❝❡s B0, . . . , Bn ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❘❛✉③②✲❱❡❡❡❝❤ ♠❛tr✐❝❡s♦r✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ❛ s❡q✉❡♥❝❡ ♦❢ ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s st❛rt✐♥❣ ❛t π ✐❢ t❤❡r❡ ❡①✐sts T = (λ, π) ❢♦r✇❤✐❝❤ Bi = Bi(T ) ❢♦r ❛❧❧ 0 ≤ i ≤ n✳ ❲❡ ✇✐❧❧ s❛② t❤❛t ❛ ♠❛tr✐① B ✐s ❛ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t ✭❛t π✮ ✐❢B = B0 · . . . ·Bn ✇❤❡r❡ B0, . . . , Bn ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❘❛✉③② ❱❡❡❝❤ ♠❛tr✐❝❡s ✭st❛rt✐♥❣ ❛t π✮✳ ❋✉rt❤❡r♠♦r❡✱✇❡ ✇✐❧❧ s❛② t❤❛t t✇♦ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝ts C,D ❝❛♥ ❜❡ ❝♦♥❝❛t❡♥❛t❡❞ ✐❢ CD ✐s ❛❧s♦ ❛ ❘❛✉③②✲❱❡❡❝❤♣r♦❞✉❝t✳

❲❡ ✇✐❧❧ s❛② t❤❛t ∆B ⊂ ∆π ✐s ❛ ❘❛✉③②✲❱❡❡❝❤ ❝②❧✐♥❞❡r ✐❢ B ✐s ❛ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t ❛t π ❛♥❞

∆B =

{(λ′, π) : λ′ =

Bλ

|Bλ|, λ ∈ ∆d

}⊂ ∆π.

❖♥❡ ❝❛♥ s❡❡ t❤❛t ❛♥② T = (λ, π) ✇❤❡r❡ λ ∈ ∆B s❛t✐s✜❡s Bi(T ) = Bi ❢♦r ❛❧❧ 0 ≤ i < n✳ ❚❤✉s✱ ∆B ✐s ❛❝②❧✐♥❞❡r s❡t ❢♦r t❤❡ s②♠❜♦❧✐❝ ❝♦❞✐♥❣ ♦❢ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥ ❣✐✈❡♥ ❜② t❤❡ s❡q✉❡♥❝❡ ♦❢ ❘❛✉③②✲❱❡❡❝❤♠❛tr✐❝❡s✳

❖♥❡ ❝❛♥ ❛♥❛❧♦❣♦✉s❧② ❞❡✜♥❡ s②♠❜♦❧✐❝ ❝②❧✐♥❞❡rs ❢♦r t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ R✳ ▲❡t ✉s ✜rst ❞❡✜♥❡ t❤❡s❡t ΘC ❛ss♦❝✐❛t❡❞ t♦ ❛ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t C = C1 · . . . · Cn st❛rt✐♥❣ ❛t π ❛♥❞ ❡♥❞✐♥❣ ❛t π′ t♦ ❜❡ t❤❡s✉❜s❡t ♦❢ s✉s♣❡♥s✐♦♥ ❞❛t❛ τ ∈ Θπ′ ✐♠♣❧✐❝✐t❡❧② ❞❡✜♥❡❞ ❜②

BTΘC = Θπ.

■♥ ♦t❤❡r ✇♦r❞s✱ ✐❢ (τ, π′, λ) ∈ X ❜❡❧♦♥❣s t♦ Θπ ×∆C ✱ t❤❡ ♣❛st n ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s ❛r❡ ♣r❡s❝r✐❜❡❞❜② C✱ ✐✳❡✳ ❢♦r −n ≤ i ≤ −1 ✇❡ ❤❛✈❡ Bi(τ, π

′, λ) = Ci+n+1✳

❈②❧✐♥❞❡rs ✐♥ t❤❡ s♣❛❝❡ X ❤❛✈❡ t❤❡♥ t❤❡ ❢♦r♠ ΘC×∆D∩X✱ ✇❤❡r❡ C ❛♥❞ D ❛r❡ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝tst❤❛t ❝❛♥ ❜❡ ❝♦♥❝❛t❡♥❛t❡❞✳ ▲❡t ✉s r❡♠❛r❦ t❤❛t ✈❡❝t♦rs τ ∈ Θπ ❛r❡ ♥♦t ♥♦r♠❛❧✐③❡❞✱ ✇❤✐❧❡ ♣♦✐♥ts (τ, π, λ)

✐♥ X ❛r❡ s✉❝❤ t❤❛t t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ✭✷✳✻✮ ❤♦❧❞s✳ ❚❤✉s✱ ∆D ×ΘC ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ X ❛♥❞

t♦ ♦❜t❛✐♥ ❛ ❝②❧✐♥❞❡r ❢♦r R ♦♥❡ ♥❡❡❞s t♦ ✐♥t❡rs❡❝t ✐t ✇✐t❤ X✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ♥♦t❛t✐♦♥

(ΘC ×∆D)(1)

+ (ΘC ×∆D) ∩ X

❢♦r ❝②❧✐♥❞❡rs t♦ ❛✈♦✐❞ ❡①♣❧✐❝✐t❧② ✇r✐t✐♥❣ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ X✳■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥s t❤❛t ✐❢ C = C1 · . . . · Cn ❛♥❞ D = D0 · . . . · Dm ✇❤❡r❡ C1, . . . , Cn✱

D0, D1, . . . , Dm ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s✱ (π, λ, τ) ∈ ΘC ×∆D ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦❝②❝❧❡♠❛tr✐❝❡s Bi = Bi(π, λ, τ) ❛s i r❛♥❣❡s ❢r♦♠ −m t♦ n ❛r❡ ✐♥ ♦r❞❡r C1, . . . , Cn, D0, D1 . . . Dm ✭✐♥ ♦t❤❡r

✇♦r❞s✱ Bi = Di ❢♦r 0 ≤ i ≤ m ❛♥❞ Bi = Ci+n+1 ❢♦r −n ≤ i ≤ −1✮✱ t❤✉s (ΘC × ∆D)(1) ❛r❡ ✐♥❞❡❡❞

s②♠❜♦❧✐❝ ❝②❧✐♥❞❡rs ❢♦r t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥✳❘❡♠❛r❦ ❛❧s♦ t❤❛t ❜② ❞❡✜♥✐t✐♦♥ ✇❡ ❤❛✈❡

✭✷✳✼✮ (ΘC ×∆π′)(1) = R−n(Θπ ×∆C)

(1).

❍✐❧❜❡rt ❞✐st❛♥❝❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❞✐❛♠❡t❡r✳ ▲❡t ✉s s❛② t❤❛t ❛ ♠❛tr✐① C ✐s ♣♦s✐t✐✈❡ ✭r❡s♣✳ ♥♦♥ ♥❡❣❛t✐✈❡✮❛♥❞ ❧❡t ✉s ✇r✐t❡ C > 0 ✭r❡s♣✳ C ≥ 0✮ ✐❢ ❛❧❧ ✐ts ❡♥tr✐❡s ❛r❡ str✐❝t❧② ♣♦s✐t✐✈❡ ✭r❡s♣✳ ♥♦♥ ♥❡❣❛t✐✈❡✮✳

❈♦♥s✐❞❡r ♦♥ t❤❡ s✐♠♣❧❡① ∆d t❤❡ ❍✐❧❜❡rt ❞✐st❛♥❝❡ dH ✱ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳

dH(λ, λ′) + log

maxi=1,...,dλiλ′i

mini=1,...,dλiλ′i

.

❖♥❡ ❝❛♥ s❡❡ t❤❛t ❢♦r ❛♥② ♥❡❣❛t✐✈❡ d×d ♠❛tr✐① A ≥ 0✱ t❤❡ ❛ss♦❝✐❛t❡❞ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥ ψA(λ) =

Aλ/|Aλ| ♦❢ ∆d ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ♦❢ t❤❡ ❍✐❧❜❡rt ❞✐st❛♥❝❡✱ ✐✳❡✳ dH(Aλ, Aλ′) ≤ dH(λ, λ

′) ❢♦r ❛♥② λ, λ′ ∈ ∆d✳❋✉rt❤❡r♠♦r❡✱ ✐❢ A > 0✱ t❤❡♥ ✐t ✐s ❛ str✐❝t ❝♦♥tr❛❝t✐♦♥✳

▲❡t ✉s ❞❡✜♥❡ t❤❡ ♣r♦❥❡❝t✐✈❡ ❞✐❛♠❛t❡r diamH(A) ♦❢ A ≥ 0 ❛s t❤❡ ❞✐❛♠❡t❡r ✇✐t❤ r❡s♣❡❝t t♦ dH ♦❢ t❤❡✐♠❛❣❡ ♦❢ ψA✱ ♥❛♠❡❧②

✭✷✳✽✮ diamH(A) + supλ,λ′∈∆d

dH(ψA(λ), ψA(λ′)) = sup

λ,λ′∈∆d

dH(Aλ,Aλ′),

✇❤❡r❡ t❤❡ ❧❛st ❡q✉❛❧✐t② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ dH ✳

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✶✺

❘❡♠❛r❦ ✷✳✻✳ ❲❡ r❡♠❛r❦ t❤❛t diamH(A) ✐s ✜♥✐t❡ ❡①❛❝t❧② ✇❤❡♥ A ✐s ❛ ♣♦s✐t✐✈❡ ♠❛tr✐①✱ s✐♥❝❡ A > 0✐s ❡q✉✐✈❛❧❡♥t t♦ ψA (∆d) ❜❡✐♥❣ ♣r❡✲❝♦♠♣❛❝t✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ✐ts ❝❧♦s✉r❡ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ∆d ❛♥❞ ✐ts❞✐❛♠❡t❡r ✇✐t❤ r❡s♣❡❝t t♦ dH ✐s ✜♥✐t❡✳

❘❡♠❛r❦ ✷✳✼✳ ◆♦t✐❝❡ t❤❛t ✐❢✱ ❣✐✈❡♥ t✇♦ ♣♦s✐t✐✈❡ ♠❛tr✐❝❡s A,B✱ ✐❢ ∆B ⊂ ∆A t❤❡♥ ❝❧❡❛r❧② ❢r♦♠ t❤❡❞❡✜♥✐t✐♦♥ diamH(B) ≤ diamH(A)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ s✐♥❝❡ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ❝②❧✐♥❞❡rs ∆AB ⊂ ∆A✱ ✇❡❤❛✈❡ t❤❛t diamH(AB) ≤ diamH(A)✳ ❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ ∆AB ✐s t❤❡ ✐♠❛❣❡ ♦❢ ∆B ❜② t❤❡ ♣r♦❥❡❝t✐✈❡tr❛♥s❢♦r♠❛t✐♦♥ ψA(λ) = Aλ/|Aλ| ✇❤✐❝❤ ✐s ❛ ♣r♦❥❡❝t✐✈❡ ❝♦♥tr❛❝t✐♦♥✱ ✇❡ ❛❧s♦ ❤❛✈❡ t❤❛t diamH(AB) ≤diamH(B)✳

✷✳✻✳ ❘❛✉③②✲❱❡❡❝❤ ❛❝❝❡❧❡r❛t✐♦♥s✳ ▲❡t T (n) + Rn(T )✱ n ≥ 0✱ ❜❡ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♦r❜✐t ♦❢ T = T (0)

s❛t✐s❢②✐♥❣ t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥✳

❆❝❝❡❧❡r❛t✐♦♥s ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♠❛♣✳ ●✐✈❡♥ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✱ ✇❡

❝❛♥ ❝♦♥s✐❞❡r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛❝❝❡❧❡r❛t✐♦♥ R ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♠❛♣✱ ❞❡✜♥❡❞ ♦♥ {T (nℓ) : ℓ ∈ N} ❜②

R(T (nℓ)) = T (nℓ+1)✱ ℓ ∈ N✳ ■♥ ♦t❤❡r ✇♦r❞s✱ R(ℓ)(T ) = R(nℓ)(T )✱ ℓ ∈ N✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ {nℓ}ℓ∈N ❛s ❛s❡q✉❡♥❝❡ ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s ❢♦r T ✳

❚❤❡ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ❝❛♥ ❜❡ ❝❤♦s❡♥ ❡✳❣✳ ❜② ❝♦♥s✐❞❡r✐♥❣ P♦✐♥❝❛ré ✜rst r❡t✉r♥ ♠❛♣ ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤✐♥❞✉❝t✐♦♥ ❛s ❢♦❧❧♦✇s✳ ❋✐① ❛ s✉❜s❡t Y ⊂ X = ∆d×R(π) ♦❢ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡✳ ❇② t❤❡ ❡r❣♦❞✐❝✐t② ♦❢ R✱ ❢♦rLeb∆d ❛❧♠♦st ❡✈❡r② λ ❛♥❞ ❢♦r π′ ∈ R(π)✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ■❊❚ T = (λ, π′) ✈✐s✐ts Y ✉♥❞❡r R ✐♥✜♥✐t❡❧②♦❢t❡♥ ❛♥❞ t❤✐s ❣✐✈❡s ✉s ✐♠♠❡❞✐❛t❡❧② ❛ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ❢♦r ❛ t②♣✐❝❛❧ T ✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛❝❝❡❧❡r❛t✐♦♥♦❢ R ✐♥ t❤✐s ❝❛s❡ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② RY ❛♥❞ ✐s ❛ ♠❛♣ RY : Y → Y ❞❡✜♥❡❞ ❛✳❡✳ ♦♥ Y ✳

❘❡♠❛r❦ ✷✳✽✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t Y = ∆B ✐s ❛ ❘❛✉③②✲❱❡❡❝❤ ❝②❧✐♥❞❡r✳ ❖♥❡ ❝❛♥ s❡❡ t❤❛t RY ✐s ♣✐❡❝❡✇✐s❡❞❡✜♥❡❞ ❛♥❞ ❧♦❝❛❧❧② ❣✐✈❡♥ ❜② ♠❛♣s ♦❢ t❤❡ ❢♦r♠ λ 7→ Dλ/|Dλ| ✇❤❡r❡ D ✐s ❛ ♠❛tr✐① ♦❢ t❤❡ ❢♦r♠ D = BC❢♦r s♦♠❡ ♥♦♥✲♥❡❣❛t✐✈❡ C ∈ SL(d,Z)✳ ❚❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ❛ ♠❛♣ ♦❢ t❤✐s ❢♦r♠ ✐s JD(λ) = |Dλ|−d ✭s❡❡❱❡❡❝❤ ❬✹✹❪✱ Pr♦♣♦s✐t✐♦♥ ✺✳✷✮✳

●✐✈❡♥ ❛♥ ❛❝❝❡❧❡r❛t✐♦♥ RY ♦♥❡ ❝❛♥ ❝♦rr❡s♣♦♥❞✐♥❣❧② ❞❡✜♥❡ ❛ ❝♦❝②❝❧❡ AY ♦✈❡r RY ♦❜t❛✐♥❡❞ ❜② ❛❝❝❡❧✲❡r❛t✐♥❣ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❝♦❝②❝❧❡✳ ❚❤✐s ❝♦❝②❝❧❡ ✐s ❛✳❡✳ ❞❡✜♥❡❞ ❜② s❡tt✐♥❣ AY (T ) + B(nY (T ))(T )✱ ✇❤❡r❡nY (T ) ✐s t❤❡ ✜rst r❡t✉r♥ t✐♠❡ ♦❢ T t♦ Y ✳

❆❝❝❡❧❡r❛t✐♦♥s ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥✳ ❙✐♠✐❧❛r❧②✱ ♦♥❡ ❝❛♥ ❛❝❝❡❧❡r❛t❡ t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥

R ♦❢ R✳ ●✐✈❡♥ T ∈ X ❛♥❞ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✱✹ ✇❡ ❞❡✜♥❡˜R ♦♥

{T (nℓ) : ℓ ∈ N} ❜②˜R(T (nℓ)) = T (nℓ+1)✱ ℓ ∈ N✳ ■♥ ♦t❤❡r ✇♦r❞s✱

˜R(ℓ)(T ) = R(nℓ)(T )✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

❛❝❝❡❧❡r❛t❡❞ ❝♦❝②❝❧❡ A = A(T ) ✐s ❣✐✈❡♥ ❜② A(ℓ,ℓ+1)(T ) + B(nℓ,nℓ+1)(T )✱ ℓ ∈ N✱ ✇❤❡r❡ B ✐s t❤❡ ❝♦❝②❝❧❡

❛ss♦❝✐❛t❡❞ t♦ R✳❆s ❜❡❢♦r❡✱ t❤❡ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ❝❛♥ ❜❡ ❝❤♦s❡♥ ❡✳❣✳ ❜② ❝♦♥s✐❞❡r✐♥❣ P♦✐♥❝❛ré ✜rst r❡t✉r♥ ♠❛♣ ♦❢ ♥❛t✉r❛❧

❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥✿ ❣✐✈❡♥ ❛ s✉❜s❡t Y ⊂ X ♦❢ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡✱ {nℓ}ℓ∈N ✐s ❞❡✜♥❡❞

❛s t❤❡ s❡q✉❡♥❝❡ ♦❢ ✈✐s✐ts ♦❢ T t♦ Y ✉♥❞❡r R ✭❛♥❞ ✐t ✐s ✇❡❧❧ ❞❡✜♥❡❞ ❢♦r ❛ t②♣✐❝❛❧ T ✮✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

❛❝❝❡❧❡r❛t✐♦♥ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② RY ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛❝❝❡❧❡r❛t❡❞ ❝♦❝②❝❧❡ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② AY❛♥❞ ✐s ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ❜②

✭✷✳✾✮ AY (T ) + B(nY (T ))(T ),

✇❤❡r❡ nY (T ) ✐s t❤❡ ✜rst r❡t✉r♥ t✐♠❡ ♦❢ T t♦ Y ✭t❤✐s ✐s ✇❡❧❧ ❞❡✜♥❡❞ ❢♦r µR❛❧♠♦st ❡✈❡r② T ∈ Y ✮✳

❉❡✜♥✐t✐♦♥ ✷✳✹✳ ▲❡t ✉s s❛② t❤❛t ❛♥ ❛❝❝❡❧❡r❛t✐♦♥ RY ♦❢ t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ R ✐s ❛ ❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r✲

❛t✐♦♥ ✐❢ Y ✐s ❛ ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❘❛✉③②✲❱❡❡❝❤ ❝②❧✐♥❞❡rs ❢♦r t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ R✳

❘❡♠❛r❦ ✷✳✾✳ ▲❡t T = (λ, π) ❜❡ ❛♥ ■❊❚ ❛♥❞ (τ, λ, π) ❛♥② ♦❢ ✐ts ❧✐❢ts ✐♥ p−1(π, λ) ∈ X✳ ■❢ RY ✐s ❛

❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥✱ t❤❡ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ♦❢ ✜rst r❡t✉r♥s t♦ Y ♦❢ t❤❡ ♦r❜✐t ♦❢ (π, λ, τ) ✉♥❞❡r R

❞❡♣❡♥❞s ♦♥ T ♦♥❧②✱ ❛♣❛rt ❢r♦♠ ♣♦ss✐❜❧② ✜♥✐t❡❧② ♠❛♥② ✐♥✐t✐❛❧ t❡r♠s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ ❡❛❝❤ ❝②❧✐♥❞❡r∆Fi × ΘEi ✐♥ Y ✐s s✉❝❤ t❤❛t Ei ✐s ♣r♦❞✉❝t ♦❢ ❛t ♠♦st ℓ0 ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s✱ t❤❡♥ nℓ ❢♦r ℓ ≥ ℓ0

✹■❢ ✇❡ ✇❛♥t t♦ ❛❝❝❡❧❡r❛t❡ ❛❧s♦ t❤❡ ❜❛❝❦✇❛r❞ ✐t❡r❛t✐♦♥s ♦❢ R✱ ✇❡ ♥❡❡❞ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs✱ ✐♥❞❡①❡❞ ❜② Z✳

✶✻ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② T ✳ ■♥❞❡❡❞ t❤❡ s❡q✉❡♥❝❡ Bn = B(Rn(π, λ, τ)) ♦❢ ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s ❢♦rn ∈ N ❞❡♣❡♥❞s ♦♥ T ♦♥❧② s✐♥❝❡ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ❡①t❡♥❞❡❞ ❘❛✉③②✲❱❡❡❝❤ ❝♦❝②❝❧❡ ❛♥❞ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥

B(Rn(π, λ, τ)) = B(Rn(π, λ))✳ ❘❡♠❛r❦ ♥♦✇ t❤❛t t♦ ❞❡❝✐❞❡ ✇❤❡t❤❡r Rnℓ(π, λ, τ) ❜❡❧♦♥❣s t♦ Y ✱ ❜②❞❡✜♥✐t✐♦♥ ♦❢ ℓ0 ❛s ♠❛①✐♠❛❧ ❝②❧✐♥❞r✐❝❛❧ ❧❡♥❣❤t✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ❦♥♦✇ t❤❡ ♠❛tr✐❝❡s Bnℓ−ℓ0 , · · ·Bnℓ+ℓ0 ❛♥❞✐❢ ℓ ≥ ℓ0✱ s✐♥❝❡ nℓ ≥ ℓ ≥ ℓ0✱ t❤❡s❡ ♠❛tr✐❝❡s ❛r❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② T ✳

✷✳✼✳ P♦s✐t✐✈✐t②✱ ❜❛❧❛♥❝❡✱ ♣r❡✲❝♦♠♣❛❝t♥❡ss ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ t❛✐❧s✳ ▲❡t T (n) + Rn(T )✱ n ≥ 0✱ ❜❡

t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♦r❜✐t ♦❢ T = T (0) s❛t✐s❢②✐♥❣ t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✷✳✺ ✭P♦s✐t✐✈❡ t✐♠❡s✮✳ ❙❡q✉❡♥❝❡ {nℓ}ℓ∈N ✐s ❝❛❧❧❡❞ ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s ❢♦r

T ✐❢ ❢♦r ❛♥② ℓ ∈ N ❛❧❧ ❡♥tr✐❡s ♦❢ B(nℓ,nℓ+1) = B(nℓ,nℓ+1)(T ) ❛r❡ str✐❝t❧② ♣♦s✐t✐✈❡✿ B(nℓ,nℓ+1) > 0✳

❘❡♠❛r❦ ✷✳✶✵✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ✭✷✳✸✮ t❤❛t ❛❧♦♥❣ ❛ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ♦❢ ♣♦s✐t✐✈❡ t✐♠❡s✱ ✇❡ ❤❛✈❡ λ(nℓ) ≥dkλ(nℓ+k)✱ ℓ ≥ 1✳

❉❡✜♥✐t✐♦♥ ✷✳✻ ✭❇❛❧❛♥❝❡❞ t✐♠❡s✮✳ ■❢✱ ❢♦r s♦♠❡ ℓ ≥ 1 ❛♥❞ ν > 0✱ ✇❡ ❤❛✈❡

✭✷✳✶✵✮1

ν≤λ(nℓ)α

λ(nℓ)β

≤ ν,1

ν≤h(nℓ)α

h(nℓ)β

≤ ν, ∀α, β ∈ A,

✇❡ s❛② t❤❛t nℓ ✐s ν✲❜❛❧❛♥❝❡❞✳ ■❢ nℓ ✐s ν✲❜❛❧❛♥❝❡❞ ❢♦r ❡❛❝❤ ℓ ≥ 1 ✭✇✐t❤ t❤❡ s❛♠❡ ν > 0✮✱ ✇❡ s❛② t❤❛t{nℓ}ℓ∈N ✐s ❛ ν✲❜❛❧❛♥❝❡❞ ✭♦r s✐♠♣❧② ❛ ❜❛❧❛♥❝❡❞✮ s❡q✉❡♥❝❡ ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s ❢♦r T ✳

❘❡♠❛r❦ ✷✳✶✶✳ ■❢ nℓ ✐s ν✲❜❛❧❛♥❝❡❞ t❤❡♥ ❧❡♥❣t❤s ❛♥❞ ❤❡✐❣❤ts ♦❢ t❤❡ ❘♦❤❧✐♥ t♦✇❡rs ❛r❡ ❛♣♣r♦①✐♠❛t❡❧② ♦❢t❤❡ s❛♠❡ s✐③❡✱ ✐✳❡✳

1

dνλ(nℓ) ≤ λ(nℓ)α ≤ λ(nℓ) ❛♥❞

1

νλ(nℓ)≤ h(nℓ)α ≤

ν

λ(nℓ)❢♦r ❡❛❝❤ α ∈ A.

Pr❡✲❝♦♠♣❛❝t♥❡ss✱ ❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥s ❛♥❞ ❜♦✉♥❞❡❞ ❞✐st♦rs✐♦♥✳ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r ❛ s♣❡❝✐❛❧ ❝❧❛ss ♦❢❛❝❝❡❧❡r❛t✐♦♥s✱ ✇❤✐❝❤ ❛r❡ ❝②❧✐♥❞r✐❝❛❧ ❛♥❞ ♣r❡❝♦♠♣❛❝t ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✳

❉❡✜♥✐t✐♦♥ ✷✳✼✳ ❲❡ s❛② t❤❛t ❛♥ ❛❝❝❡❧❡r❛t✐♦♥ RY ♦❢ R ✐s ♣r❡✲❝♦♠♣❛❝t ✇❤❡♥❡✈❡r Y ✐s ♣r❡✲❝♦♠♣❛❝t ✐♥ X

❛♥❞ ✇❡ s❛② t❤❛t ❛♥ ❛❝❝❡❧❡r❛t✐♦♥ RY ♦❢ t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ R ✐s ♣r❡✲❝♦♠♣❛❝t ✇❤❡♥❡✈❡r Y ✐s ♣r❡✲❝♦♠♣❛❝t

✐♥ X✳

❆ ❘❛✉③②✲❱❡❡❝❤ ❝②❧✐♥❞❡r ∆C ✐s ♣r❡✲❝♦♠♣❛❝t ✐♥ X ✐❢ ❛♥❞ ♦♥❧② ✐❢ C ✐s ❛ ♣♦s✐t✐✈❡ ♠❛tr✐①✳ ❚❤✉s✱ ❛❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♣r❡❝♦♠♣❛❝t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ t❤❡ ❝②❧✐♥❞❡rs ✐♥ t❤❡ ✜♥✐t❡ ✉♥✐♦♥ ❞❡✜♥✐♥❣t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❛ ♣♦s✐t✐✈❡ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t✳ ❙✐♠✐❧❛r❧②✱ t❤❡r❡ ❛r❡ s✐♠♣❧❡ ❝♦♥❞✐t✐♦♥s ♦♥

❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t ♠❛tr✐❝❡s C,D t❤❛t ❣✉❛r❛♥t❡❡ t❤❛t ❛ ❝②❧✐♥❞❡r ∆D ×ΘC ❢♦r R ✐s ♣r❡✲❝♦♠♣❛❝t ✭s❡❡❢♦r ❡①❛♠♣❧❡ t❤❡ ♥♦t✐♦♥ ♦❢ str♦♥❣❧② ♣♦s✐t✐✈❡ ♠❛tr✐① ✐♥ ❬✸❪✮✳

❘❡♠❛r❦ ✷✳✶✷ ✭s❡❡✱ ❡✳❣✳✱ ❬✹✶❪✮✳ ■❢ RY ✐s ❛ ♣r❡✲❝♦♠♣❛❝t ❛❝❝❡❧❡r❛t✐♦♥ t❤❡♥ ❢♦r ❛♥② T ✱ ❢♦r ✇❤✐❝❤ t❤❡❝♦rr❡s♣♦♥❞✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s {nℓ}ℓ∈N ✐s ✇❡❧❧✲❞❡✜♥❡❞✱ {nℓ}ℓ∈N ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❜❛❧❛♥❝❡❞✳

❋✉rt❤❡r♠♦r❡✱ ✐❢ t❤❡ ♣r❡✲❝♦♠♣❛❝t ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝②❧✐♥❞r✐❝❛❧ ✭✐✳❡✳ t❤❡ ✐♥❞✉❝✐♥❣ s❡t Y ✐s ❛ ✜♥✐t❡ ✉♥✐♦♥ ♦❢

♣r❡✲❝♦♠♣❛❝t ❝②❧✐♥❞❡rs ❢♦r R✮✱ t❤❡♥ ❡❛❝❤ r❡s✉❧t✐♥❣ {nℓ}ℓ∈N ✐s ❛✉t♦♠❛t✐❝❛❧❧② ♣♦s✐t✐✈❡ ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

T ✳

❖♥❡ ♦❢ t❤❡ r❡❛s♦♥ ✇❤② ♣r❡✲❝♦♠♣❛❝t ❛❝❝❡❧❡r❛t✐♦♥s ❛r❡ ✐♠♣♦rt❛♥t ✐s t❤❛t t❤❡② ❡♥❥♦② t❤❡ ❜♦✉♥❞❡❞❞✐st♦rs✐♦♥ ♣r♦♣❡rt②✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ ✇❡ s❡t Y = ∆C ✇❤❡r❡ C ✐s ❛ ♣♦s✐t✐✈❡ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t ❛♥❞❝♦♥s✐❞❡r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥✱ RY ✐s str✐❝t❧② ❡①♣❛♥❞✐♥❣ ❛♥❞ ❤❛s ❜♦✉♥❞❡❞ ❞✐st♦rs✐♦♥✱✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t νY s✉❝❤ ❢♦r ❛♥② ✐♥✈❡rs❡ ❜r❛♥❝❤ ♦❢ RY ✱ ✇❤✐❝❤ ✐s ❛ ♠❛♣ ♦❢ t❤❡ ❢♦r♠ λ 7→Dλ/|Dλ|✱ ✇❤❡r❡ D ✐s ❛ ♠❛tr✐① ♦❢ t❤❡ ❢♦r♠ ❞❡s❝r✐❜❡❞ ✐♥ ❘❡♠❛r❦ ✷✳✽✱ t❤❡ ❏❛❝♦❜✐❛♥ JD(λ) ♦❢ t❤❡ ✐♥✈❡rs❡❜r❛♥❝❤ s❛t✐s✜❡s |JD(λ)|/|JD(λ

′)| ≤ νY ❢♦r ❛❧❧ λ, λ′ ∈ Y ✳ ❚❤✐s ♣r♦♣❡rt② ❢♦❧❧♦✇s ❢r♦♠ ❛ r❡♠❛r❦ ❜② ❱❡❡❝❤✭s❡❡ ❬✹✹❪✱ ❙❡❝t✐♦♥ ✺✮ ❛♥❞ ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r ❡①❛♠♣❧❡ ✐♥ ❬✸✵❪ ✭s❡❡ ▲❡♠♠❛ ✸✳✹✮ ♦r ❬✸❪ ✭s❡❡ ▲❡♠♠❛ ✹✳✹✮✳

❚♦ ❝♦♥tr♦❧ ❞✐st♦rs✐♦♥✱ ✐t ✐s ✉s❡❢✉❧ t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t② ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❙❡❝t✐♦♥ ✷ ✐♥❬✺❪✮✳ ●✐✈❡♥ ❛ d× d ♣♦s✐t✈❡ ♠❛tr✐① C✱ ❧❡t ✉s ❞❡✜♥❡ νcol(C) t♦ ❜❡

✭✷✳✶✶✮ νcol(C) + max1≤i,j,k≤d

CijCik

.

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✶✼

▲❡t ✉s r❡♠❛r❦ t❤❛t ✭s❡❡ ❛❧s♦ Pr♦♣♦s✐t✐♦♥ ✷ ✐♥ ❬✺❪✮ ✐❢ C ✐s ❛ d× d ♠❛tr✐① ✇✐t❤ ♥♦♥ ♥❡❣❛t✐✈❡ ❡♥tr✐❡s ❛♥❞D ✐s ❛ d× d ♠❛tr✐① ✇✐t❤ ♣♦s✐t✐✈❡ ❡♥tr✐❡s ✭s♦ t❤❛t ✐♥ ♣❛rt✐❝✉❧❛r CD ❤❛s ♣♦s✐t✐✈❡ ❡♥tr✐❡s ❛♥❞ νcol(CD) ✐s✇❡❧❧ ❞❡✜♥❡❞✮ ♦♥❡ ❤❛s

✭✷✳✶✷✮ νcol(CD) ≤ νcol(D).

❚❤❡♥ ♦♥❡ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ✭s❡❡ ❛❧s♦ ❈♦r♦❧❧❛r② ✶✳✼ ✐♥ ❬✶✽❪ ❛♥❞ ❡q✉❛t✐♦♥ ✭✶✺✮ ✐♥ ❬✺❪✮✳

▲❡♠♠❛ ✷✳✶✸ ✭❞✐st♦rs✐♦♥✮✳ ■❢ C ✐s ❛ d× d ♣♦s✐t✐✈❡ ♠❛tr✐①✱ t❤❡ ❏❛❝♦❜✐❛♥ JC ♦❢ t❤❡ ♠❛♣ λ 7→ Cλ/|Cλ|s❛t✐s✜❡s

supλ,λ′∈∆d

∣∣∣∣JC(λ)

JC(λ′)

∣∣∣∣ ≤ νcol(C)d.

Pr♦♦❢✳ ❙✐♥❝❡ ∆d ✐s ❣❡♥❡r❛t❡❞ ❜② ✐ts ✈❡rt✐❝❡s✱ ✇❤♦s❡ ✐♠❛❣❡ ✉♥❞❡r t❤❡ ♠❛♣ λ 7→ Cλ ❛r❡ t❤❡ ❝♦❧✉♠♥s ♦❢C✱ ✇❡ ❤❛✈❡

supλ,λ′∈∆d

|Cλ′|

|Cλ|= max

1≤j,k≤d

∑di=1Cij∑di=1Cik

≤ max1≤j,k≤d

∑di=1

(CijCik

)Cik

∑di=1Cik

≤ νcol(C).

❚❤✉s✱ t❤❡ ❡st✐♠❛t❡ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠ ♦❢ JC(λ) ✭s❡❡ ❘❡♠❛r❦ ✷✳✽✮✳ �

❊①♣♦♥❡♥t✐❛❧ t❛✐❧s✳ ❚❤❡ ♠❛✐♥ t❡❝❤♥✐❝❛❧ t♦♦❧ ❢♦r ✉s ✐s t❤❡ r❡s✉❧t ♣r♦✈❡❞ ❜② ❆✈✐❧❛✱ ●♦✉ë③❡❧ ❛♥❞ ❨♦❝❝♦③ ✐♥❬✸❪ ✭✐♥ ♦r❞❡r t♦ s❤♦✇ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ♦❢ t❤❡ ❚❡✐❝❤♠✉❡❧❧❡r ✢♦✇✮✱ ✐✳❡✳ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♣r❡✲❝♦♠♣❛❝t❛❝❝❡❧❡r❛t✐♦♥s ❢♦r ✇❤✐❝❤ t❤❡ r❡t✉r♥ t✐♠❡ ❤❛s ❡①♣♦♥❡♥t✐❛❧ t❛✐❧s✳ ❯s✐♥❣ t❤❡ t❡r♠✐♥♦❧♦❣② ✐♥tr♦❞✉❝❡❞ s♦ ❢❛r✱♦♥❡ ❝❛♥ r❡♣❤r❛s❡ t❤❡ ♠❛✐♥ r❡s✉❧t ♣r♦✈❡❞ ✐♥ ❬✸❪ ❛s ❢♦❧❧♦✇s✳

❚❤❡♦r❡♠ ✷✳✶✹ ✭❚❤❡♦r❡♠ ✹✳✶✵ ✐♥ ❬✸❪✮✳ ❋♦r ❡✈❡r② δ > 0✱ t❤❡r❡ ❡①✐sts ❛ ❝②❧✐♥❞r✐❝❛❧ ♣r❡✲❝♦♠♣❛❝t ❛❝❝❡❧❡r✲

❛t✐♦♥ RYδ✭❝♦rr❡s♣♦♥❞✐♥❣ t♦ r❡t✉r♥s t♦ ❛ s❡t Yδ ⊂ X ✇❤✐❝❤ ✐s ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❝②❧✐♥❞❡rs ❢♦r R✮ s✉❝❤ t❤❛t

t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛❝❝❡❧❡r❛t❡❞ ❝♦❝②❝❧❡ AYδ ❣✐✈❡♥ ❜② ✭✷✳✾✮ s❛t✐s✜❡s

✭✷✳✶✸✮

∫

Yδ

‖AYδ(T )‖1−δ dµ

R(T ) <∞.

◆♦t❡ t❤❛t ❜② ❘❡♠❛r❦ ✷✳✶✷ ✇❡ ✐♠♠❡❞✐❛t❡❧② ♦❜t❛✐♥ t❤❛t t❤❡ t✐♠❡s ✐♥ t❤❡ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ❝♦rr❡s♣♦♥❞✲

✐♥❣ t♦ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ RYδ❛r❡ ♣♦s✐t✐✈❡ ❛♥❞ ❜❛❧❛♥❝❡❞✳ ❚❤❡ ♦r✐❣✐♥❛❧ st❛t❡♠❡♥t ♦❢ ❚❤❡♦r❡♠ 4.10 ✐♥ ❬✸❪

❝❧❛✐♠s t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ♦❢ e(1−δ)r

Yδ(T )

✱ ✇❤❡r❡ rYδ(T ) ✐s t❤❡ ✜rst r❡t✉r♥ t✐♠❡ ♦❢ T ∈ Yδ ✉♥❞❡r t❤❡ ❱❡❡❝❤

✢♦✇✳ ❋♦r t❤❡ r❡❞✉❝t✐♦♥ t♦ t❤✐s ❢♦r♠✉❧❛t✐♦♥✱ s❡❡ ❬✹✶❪ ❛♥❞ r❡❝❛❧❧ t❤❡ ♥♦t❛t✐♦♥ ✐♥tr♦❞✉❝❡❞ ❛❜♦✈❡✳ ❲❡ r❡❝❛❧❧t❤❛t ❇✉❢❡t♦✈✱ ❜② ❞✐✛❡r❡♥t t❡❝❤♥✐q✉❡s✱ ♦❜t❛✐♥❡❞ ✐♥ ❬✺❪ ❛ r❡s✉❧t ❛♥❛❧♦❣♦✉s t♦ ✭✷✳✶✸✮ ❢♦r s♦♠❡ δ ∈ (0, 1)✳❲❡ ✇✐❧❧ ♥❡❡❞ t❤❡ ❢✉❧❧ str❡♥❣❤t ♦❢ t❤❡ r❡s✉❧t ♦❢ ❬✸❪✱ ✐✳❡✳ ❢♦r ❛♥② δ ∈ (0, 1)✳

✸✳ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ❢♦r ■❊❚s

❆s ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✱ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ❢♦r ■❊❚s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡❣r♦✇t❤ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❝♦❝②❝❧❡ ♠❛tr✐❝❡s ❛❧♦♥❣ ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ t✐♠❡s✳ ■♥ t❤✐s s❡❝t✐♦♥✱✇❡ ✜rst ❞❡✜♥❡ ❛ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ❢♦r ■❊❚s ✭✐♥ ❉❡✜♥✐t✐♦♥ ✸✳✶✮ ✇❤✐❝❤ ❤♦❧❞s ❢♦r ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t ♦❢■❊❚s ❛♥❞ t❤❛t ✇❛s ✉s❡❞ ❜② t❤❡ t❤✐r❞ ❛✉t❤♦r ✐♥ ❬✹✶❪ ❛♥❞ ❜② ❘❛✈♦tt✐ ✐♥ ❬✸✺❪ t♦ ♣r♦✈❡ ♠✐①✐♥❣ ❢♦r s♣❡❝✐❛❧✢♦✇s ♦✈❡r ■❊❚s ✭♠♦r❡ ♣r❡❝✐s❡❧②✱ t❤✐s ❝♦♥❞✐t✐♦♥ ❛❧❧♦✇s t♦ ♣r♦✈❡ t❤❛t t❤❡ ❇✐r❦❤♦✛ s✉♠s Sr(f) ♦❢ ❛ ❢✉♥❝t✐♦♥f ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡s s❛t✐s❢② ♣r❡❝✐s❡ q✉❛♥t✐t❛t✐✈❡ ❡st✐♠❛t❡s✱ s❡❡Pr♦♣♦s✐t✐♦♥ ✹✳✹✮✳ ❲❡ t❤❡♥ ❞❡✜♥❡ ❛ str♦♥❣❡r ❉✐♦♣❤❛♥t✐♥❡ ❈♦♥❞✐t✐♦♥ ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✸✳✷ ✐♥ ❙❡❝t✐♦♥ ✸✳✷✮✇❤✐❝❤ ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ ♣r♦✈❡ ❛ q✉❛♥t✐t❛t✐✈❡ ❢♦r♠ ♦❢ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ❛♥❞✱ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ♠✐①✐♥❣ ♦❢❛❧❧ ♦r❞❡rs ❢♦r t❤❡ s❛♠❡ t②♣❡ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ✇❤♦s❡ ❜❛s❡ ■❊❚ ❡♥❥♦②s t❤✐s str♦♥❣❡r ❉✐♦♣❤❛♥t✐♥❡ ♣r♦♣❡rt②✳❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t❤❛t t❤✐s ❝♦♥❞✐t✐♦♥ ✐s s❛t✐❢s✐❡❞ ❜② ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t ♦❢ ■❊❚s ✭s❡❡Pr♦♣♦s✐t✐♦♥ ✸✳✻✮✳

✶✽ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

✸✳✶✳ ▼✐①✐♥❣ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② t❤❡t❤✐r❞ ❛✉t❤♦r ✐♥ ❬✹✶❪ t♦ s❤♦✇ ♠✐①✐♥❣ ❢♦r t❤❡ ❝❧❛ss ♦❢ s♣❡❝✐❛❧ ✢♦✇s ✉♥❞❡r ❛ r♦♦❢ ❢✉♥❝t✐♦♥ ✇✐t❤ ♦♥❧② ♦♥❡❛s②♠♠❡tr✐❝ s✐♥❣✉❧❛r✐t②✳ ❘❛✈♦tt✐ ✐♥ ❬✸✺❪ ❡①t❡♥❞s t❤✐s r❡s✉❧t ❛♥❞ s❤♦✇s t❤❛t✱ ✐♥ ❢❛❝t✱ t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥✐♠♣❧✐❡s ♠✐①✐♥❣ ❛❧s♦ ✇❤❡♥ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥ ❤❛s s❡✈❡r❛❧ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ✭s❡❡ ❚❤❡♦r❡♠✸✳✷ ❜❡❧♦✇✮✳

❉❡✜♥✐t✐♦♥ ✸✳✶ ✭▼✐①✐♥❣ ❉❈✱ s❡❡ ❬✹✶❪✮✳ ❲❡ s❛② t❤❛t ❛♥ ■❊❚ T s❛t✐s✜❡s t❤❡ ♠✐①✐♥❣ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥✭♦r✱ ❢♦r s❤♦rt✱ s❛t✐s✜❡s t❤❡ ♠✐①✐♥❣ ❉❈ ✮ ✇✐t❤ ✐♥t❡❣r❛❜✐❧✐t② ♣♦✇❡r τ ✐❢ 1 < τ < 2 ❛♥❞ t❤❡r❡ ❡①✐st ℓ ∈ N✱ν > 1 ❛♥❞ ❛ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ♦❢ ❜❛❧❛♥❝❡❞ ✐♥❞✉❝t✐♦♥ t✐♠❡s s✉❝❤ t❤❛t✿

• t❤❡ s✉❜s❡q✉❡♥❝❡ {nℓk}k∈N ✐s ♣♦s✐t✐✈❡✱

• t❤❡ ♠❛tr✐❝❡s B(nℓ,nℓ+ℓ) ❤❛✈❡ ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ❞✐❛♠❡t❡r ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❍✐❧❜❡rt ♠❡tr✐❝

✭r❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ✐♥ ✭✷✳✽✮✮✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts D > 0 s✉❝❤ t❤❛t diamH(B(nℓ,nℓ+ℓ)) ≤ D ❢♦r ❛♥②

ℓ ≥ 1✳• s❡tt✐♥❣ Aℓ + B(nℓ,nℓ+1)✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿

✭✸✳✶✮ limℓ→+∞

‖Aℓ‖

ℓτ= lim

ℓ→+∞

‖B(nℓ,nℓ+1)‖

ℓτ= 0.

❲❡ ❞❡♥♦t❡ ❜② MDC(τ, ℓ, ν

)t❤❡ s❡t ♦❢ ■❊❚s ✇❤✐❝❤ s❛t✐s❢② t❤❡ ♠✐①✐♥❣ ❉❈ ✇✐t❤ ✐♥t❡❣r❛❜✐❧✐t② ♣♦✇❡r

1 < τ < 2 ❛♥❞ ♣❛r❛♠❡t❡rs ℓ ∈ N ❛♥❞ ν > 1 ❛♥❞ ✇❡ ❞❡♥♦t❡ ❜② MDC (τ) t❤❡ ■❊❚s ✇❤✐❝❤ s❛t✐s❢② t❤❡♠✐①✐♥❣ ❉❈ ✇✐t❤ ✐♥t❡❣r❛❜✐❧✐t② ♣♦✇❡r τ ✱ t❤❛t ✐s t❤❡ ✉♥✐♦♥ ♦✈❡r ℓ ∈ N ❛♥❞ ν > 1 ♦❢ MDC

(τ, ℓ, ν

)✳ ■❢

T ∈MDC(τ, ℓ, ν

)❢♦r s♦♠❡ 1 < τ < 2✱ ℓ ∈ N ❛♥❞ ν > 1✱ ✇❡ s✐♠♣❧② s❛② t❤❛t T s❛t✐s✜❡s t❤❡ ♠✐①✐♥❣ ❉❈✳

❘❡♠❛r❦ ✸✳✶ ✭❬✹✶❪✮✳ ❲❡ r❡♠❛r❦ t❤❛t ✭✸✳✶✮ ✐♠♣❧✐❡s ❢♦r d = 2 t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✉s❡❞ ❢♦r r♦t❛t✐♦♥s✐♥ ❬✸✾❪ t♦ ♣r♦✈❡ ♠✐①✐♥❣ ❢♦r t②♣✐❝❛❧ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥ts ✐♥ ❣❡♥✉s ♦♥❡ ✭✐✳❡✳ kℓ = o(ℓτ )✱ ✇❤❡r❡ {kℓ}ℓ∈N ❛r❡t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❛♥❞ t❤❡ ❡①♣♦♥❡♥t τ s❛t✐s✜❡s t❤❡ s❛♠❡ ❛ss✉♠♣t✐♦♥ 1 < τ < 2✮✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ✇❛s ♣r♦✈❡❞ ❜② t❤❡ t❤✐r❞ ❛✉t❤♦r ✐♥ ❬✹✶❪ ✇❤❡♥ f ❤❛s ♦♥❧② ♦♥❡ s✐♥❣✉❧❛r✐t②✱ t❤❡♥❡①t❡♥❞❡❞ t♦ s❡✈❡r❛❧ s✐♥❣✉❧❛r✐t✐❡s ❜② ❘❛✈♦tt✐ ❬✸✺❪✳

❚❤❡♦r❡♠ ✸✳✷ ✭❬✹✶✱ ✸✺❪✮✳ ■❢ T : I → I s❛t✐s✜❡s t❤❡ ▼✐①✐♥❣ ❉❈✱ ❢♦r ❡✈❡r② r♦♦❢ ❢✉♥❝t✐♦♥ f : I → R+ ✇✐t❤

❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛t t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ T ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❉❡✜♥✐t✐♦♥ ✷✳✶✮✱ t❤❡s♣❡❝✐❛❧ ✢♦✇ (ϕt)t∈R ♦✈❡r T ✉♥❞❡r f ✐s ♠✐①✐♥❣✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ Pr♦♣♦s✐t✐♦♥ ✐s ♣r♦✈❡❞ ❜② t❤❡ t❤✐r❞ ❛✉t❤♦r ✐♥ ❬✹✶❪ ✭s❡❡ t❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✷✺✮✳

Pr♦♣♦s✐t✐♦♥ ✸✳✸ ✭❬✹✶❪✮✳ ❋♦r ❛♥② 1 < τ < 2✱ t❤❡r❡ ❡①✐sts ℓ ∈ N ❛♥❞ ν > 1 s✉❝❤ t❤❛t t❤❡ s❡tMDC(τ, ℓ, ν

)

❤❛s ❢✉❧❧ ♠❡❛s✉r❡✱ ✐✳❡✳ ❢♦r ❡❛❝❤ ✐rr❡❞✉❝✐❜❧❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t✉♠ π ❛♥❞ ❢♦r ▲❡❜❡s❣✉❡ ❛✳❡✳ λ ∈ ∆d✱ t❤❡❝♦rr❡s♣♦♥❞✐♥❣ ■❊❚ T = (λ, π) ❜❡❧♦♥❣s t♦ MDC

(τ, ℓ, ν

)✳

❘❡♠❛r❦ ✸✳✹✳ ❚❤❡ ❦❡② ✐♥❣r❡❞✐❡♥t ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❛❜♦✈❡ r❡s✉❧t ✐♥ ❬✹✶❪ ✐s t❤❡ ♠❛✐♥ ❡st✐♠❛t❡ ♦♥❡①♣♦♥❡♥t✐❛❧ t❛✐❧s ❢r♦♠ ❬✸❪ ✇❤✐❝❤ ✇❡ r❡❝❛❧❧ ✐♥ ❚❤❡♦r❡♠ ✷✳✶✹✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ s❡q✉❡♥❝❡ {nℓ}ℓ∈N❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♠✐①✐♥❣ ❉❈ ✐s ❝♦♥str✉❝t❡❞ ❢r♦♠ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s ❝♦rr❡s♣♦♥❞✐♥❣t♦ t❤❡ ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❝②❧✐♥❞❡rs Yδ✱ δ = 1 − τ−1 ❢r♦♠ ❚❤❡♦r❡♠ ✷✳✶✹ ✭r❡❝❛❧❧ ❢r♦♠ ❘❡♠❛r❦ ✷✳✾ t❤❛t t❤✐ss❡q✉❡♥❝❡ ❡ss❡♥t✐❛❧❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ■❊❚ ♦♥❧②✮✳ ❚❤❡ ✜rst t✇♦ ❝♦♥❞✐t✐♦♥s ✐♥ ❉❡✜♥✐t✐♦♥ ✸✳✶ ❢♦❧❧♦✇ ❡❛s✐❧②❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤✐s ❛❝❝❡❧❡r❛t✐♦♥ ✐s ❝②❧✐♥❞r✐❝❛❧ ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ ❡①♣♦♥❡♥t✐❛❧ t❛✐❧s ❝♦♥❞✐t✐♦♥ ✐♥ ❚❤❡♦r❡♠ ✷✳✶✹ ✭s❡❡ t❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✷ ✐♥ ❬✹✶❪✮✳

✸✳✷✳ ❘❛t♥❡r ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✇❤✐❝❤ ✇❡✇✐❧❧ ❧❛t❡r ✉s❡ ✭s❡❡ ✐♥ ♣❛rt✐❝✉❧❛r ✐♥ s❡❝t✐♦♥s ✹✳✸ ❛♥❞ ✺✳✷✮ t♦ q✉❛♥t✐❢② ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ❛♥❞ ♣r♦✈❡ t❤❡s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ❢♦r s✉s♣❡♥s✐♦♥ ✢♦✇s ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ❋♦r t❤✐s ✇❡✇✐❧❧ ♥❡❡❞ s♦♠❡ ♥♦t❛t✐♦♥✳ ▲❡t T ❜❡ ❛♥ ■❊❚ s❛t✐s❢②✐♥❣ t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥✳ ❘❡❝❛❧❧ t❤❛t B−1 ❞❡♥♦t❡s

✺■♥ t❤❡ st❛t❡♠❡♥t ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✷ ✐♥ ❬✹✶❪✐t ✐s ♦♥❧② ❝❧❛✐♠❡❞ t❤❛t ❢♦r 1 < τ < 2 t❤❡ s❡t ♦❢ ■❊❚s ✇❤✐❝❤ s❛t✐s✜❡s t❤❡▼✐①✐♥❣ ❉❈ ✇✐t❤ ✐♥t❡❣r❛❜✐❧✐t② ♣♦✇❡r τ ✱ ✐✳❡✳ ✇❤❛t ✇❡ ❤❡r❡ ❝❛❧❧ MDC (τ) ❤❛s ❢✉❧❧ ♠❡❛s✉r❡✳ ❇② r❡❛❞✐♥❣ t❤❡ ❛❝t✉❛❧ ♣r♦♦❢ ♦❢Pr♦♣♦s✐t✐♦♥ ✸✳✷ ✐♥ ❬✹✶❪✱ t❤♦✉❣❤✱ ♦♥❡ ❝❛♥ s❡❡ t❤❛t ℓ ∈ N ❛♥❞ ν > 1 ❛♥❞ ❝❤♦s❡♥ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ♣r♦♦❢ ❛♥❞ t❤❡ ❢✉❧❧♠❡❛s✉r❡ s❡t ♦❢ ■❊❚s ❝♦♥str✉❝t❡❞ ❛❧❧ s❤❛r❡ t❤❡ s❛♠❡ ♣❛r❛♠❡t❡rs ℓ ❛♥❞ ν✱ ✐✳❡✳ t❤❡ ♣r♦♦❢ ❞♦❡s s❤♦✇ ✐♥❞❡❡❞ t❤❛t t❤❡ r❡s✉❧t❤❡r❡ ❝✐t❡❞ ❤♦❧❞s✳

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✶✾

t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❝♦❝②❝❧❡ ❛♥❞ t❤❛t h(n) = (h(n)α )α∈A ✐s t❤❡ ✈❡❝t♦r ♦❢ ❤❡✐❣❤ts ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✺✮✳ ●✐✈❡♥ ❛

s❡q✉❡♥❝❡ {nℓ}ℓ∈N ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s✱ ✇❡ ❞❡✜♥❡

✭✸✳✷✮ Aℓ = Aℓ(T ) + B(nℓ,nℓ+1)(T ), qℓ + maxα∈A

h(nℓ)α ❢♦r ℓ ≥ 1

✭❡q✉✐✈❛❧❡♥t❧②✱ qℓ ✐s t❤❡ ♥♦r♠ ♦❢ t❤❡ ❧❛r❣❡st ❝♦❧✉♠♥ ♦❢ t❤❡ tr❛♥s♣♦s❡ ♦❢ t❤❡ ♠❛tr✐① A(ℓ) = B(nℓ)✮✳❚❤❡ ❛❜♦✈❡ ♥♦t❛t✐♦♥ ✐s ❝❤♦s❡♥ t❤✐s ✇❛② t♦ r❡s❡♠❜❧❡ t❤❡ st❛♥❞❛r❞ ♥♦t❛t✐♦♥ r❡❧❛t❡❞ t♦ t❤❡ ❝♦♥t✐♥✉❡❞❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ❛❧❣♦r✐t❤♠✱ s✐♥❝❡ Aℓ ❛♥❞ qℓ ♣❧❛② ❛♥ ❛♥❛❧♦❣♦✉s r♦❧❡ t♦ ❡♥tr✐❡s ❛♥❞ ❞❡♥♦♠✐♥❛t♦rs ♦❢ t❤❡❝♦♥✈❡r❣❡♥ts ♦❢ t❤❡ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❛❧❣♦r✐t❤♠✱ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❛❧❣♦r✐t❤♠✳ ❲❡ r❡♠❛r❦ t❤❛t s✐♥❝❡ ✇❡ ❡①t❡♥❞❡❞ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ❝♦❝②❝❧❡ B−1 t♦ ❛

❝♦❝②❝❧❡ ♦✈❡r t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ R ♦❢ R ✭s❡❡ ❙❡❝t✐♦♥ ✷✳✺✮✱ ✇❡ ❝❛♥ ❞❡✜♥❡ Aℓ ❛❧s♦ ❢♦r ♥❡❣❛t✐✈❡ ✐♥❞❡①❡sℓ✱ ❜② s❡tt✐♥❣✿

✭✸✳✸✮ Aℓ = Aℓ(T ) + B(nℓ,nℓ+1)(T ), ℓ ∈ Z, T = (τ, π, λ) ∈ X.

❋r♦♠ ♥♦✇ ♦♥ ✇❡ ❛ss✉♠❡ t❤❛t ℓ, ν ❛r❡ s♦ t❤❛t t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✸ ❤♦❧❞s✳

❉❡✜♥✐t✐♦♥ ✸✳✷ ✭❘❛t♥❡r ❉❈✮✳ ❲❡ s❛② t❤❛t ❛♥ ■❊❚ T s❛t✐s✜❡s t❤❡ ❘❛t♥❡r ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✐❢ Ts❛t✐s✜❡s t❤❡ ▼✐①✐♥❣ ❉❈ MDC

(τ, ℓ, ν

)❛❧♦♥❣ ❛ s✉❜s❡q✉❡♥❝❡ (nℓ)ℓ∈N ❢♦r s♦♠❡ ν > 1 ❛♥❞ l ∈ N ❛♥❞ t❤❡r❡

❡①✐sts ξ < 1, η < 1 s✉❝❤ t❤❛t Aℓ ❛♥❞ qℓ ❞❡✜♥❡❞ ✐♥ ✭✸✳✷✮ s❛t✐s❢②

✭✸✳✹✮∑

ℓ∈N s✳t✳ ‖Aℓ‖‖Aℓ+1‖...‖Aℓ+L‖>ℓξ

1

(log qℓ)η< +∞, ✇❤❡r❡ L = ℓ(1 +

[logd(2ν

2)]).

✭❍❡r❡ [·] ❞❡♥♦t❡s t❤❡ ✐♥t❡❣❡r ♣❛rt✮✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ s❛② t❤❛t T ∈ RDC (τ, ξ, η)✳ ❲❡ ❛❧s♦ ✇r✐t❡ RDC (τ)❢♦r t❤❡ ✉♥✐♦♥ ♦❢ ❛❧❧ RDC (τ, ξ, η) ♦✈❡r ξ < 1 ❛♥❞ η < 1✳

❲❡ r❡♠❛r❦ t❤❛t ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ RDC (τ, ξ, η) ✇❡ ❞♦ ♥♦t r❡❝♦r❞ t❤❡ ❡①♣❧✐❝✐t ❞❡♣❡♥❞❡♥❝❡ ♦♥ ν > 1❛♥❞ l ∈ N✱ ✐t ✐s s✉✣❝✐❡♥t t❤❛t t❤❡ ❘❛t♥❡r ❉❈ ❤♦❧❞s ✇✐t❤ r❡s♣❡❝t t♦ s♦♠❡ s✉❝❤ η ❛♥❞ l✳ ■♥ t❤❡ r❡st ♦❢t❤❡ ♣❛♣❡r✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ❘❛t♥❡r ❉❈ RDC (τ, ξ, η) ♦♥❧② ❢♦r ✈❛❧✉❡s 1 < τ < 16/15✳

❘❡♠❛r❦ ✸✳✺✳ ❚❤❡ ❘❛t♥❡r ❉❈ s❤♦✉❧❞ ❜❡ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ❈♦♥❞✐t✐♦♥ ✐♥tr♦❞✉❝❡❞ ❜②❋❛②❛❞ ❛♥❞ t❤❡ ✜rst ❛✉t❤♦r ✐♥ ❬✶✵❪✳ ■♥ ❬✶✵❪ t❤❡ ❛✉t❤♦rs ❞❡✜♥❡

E +

α ∈ [0, 1) :

∑

i/∈Kα

1

log7/8 qi< +∞

,

✇❤❡r❡ Kα = {i ∈ N : qi+1 < qi log7/8 qi}✱ ❛♥❞ s❤♦✇ t❤❛t λ(E) = 1 ✭s❡❡ Pr♦♣♦s✐t✐♦♥ ✶✳✼ ✐♥ ❬✶✵❪✮✳ ❚❤✐s

❝♦rr❡s♣♦♥❞s t♦ ❘❛t♥❡r ❉❈ ✇✐t❤ ξ = η = 7/8✳◆♦t✐❝❡ t❤❛t ✐❢ ❛♥ ■❊❚ T ✐s ♦❢ ❜♦✉♥❞❡❞ t②♣❡ ✭✇❤✐❝❤ ❢♦r♠ ❛ 0 ♠❡❛s✉r❡ s❡t✮ ✐✳❡✳ ❢♦r ❛❧❧ ℓ ∈ N✱

‖Aℓ(T )‖ < C✱ t❤❡♥ ❘❛t♥❡r ❉❈ ✐s ❛✉t♦♠❛t✐❝❛❧❧② s❛t✐s✜❡❞ ✭✇❡ s✉♠ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② t❡r♠s✮✳ ❘❛t♥❡r❉❈ ♠❡❛♥s t❤❛t t❤❡ t✐♠❡s ℓ✱ ✇❤❡r❡ Aℓ(T ) ✐s ❧❛r❣❡ ❛r❡ ♥♦t t♦♦ ❢r❡q✉❡♥t✳ ■♥ ❛ s❡♥s❡ ✐❢ ❛♥ ■❊❚ s❛t✐s✜❡s❘❛t♥❡r ❉❈✱ ✐t ❜❡❤❛✈❡s ❧✐❦❡ ❛♥ ■❊❚ ♦❢ ❜♦✉♥❞❡❞ t②♣❡ ♠♦❞✉❧♦ s♦♠❡ ❡rr♦r ✇✐t❤ s♠❛❧❧ ❞❡♥s✐t② ✭❛s ❛ s✉❜s❡t♦❢ N✮✱ ❜✉t✱ ❛s Pr♦♣♦s✐t✐♦♥ ✸✳✻ s❤♦✇s✱ t❤✐s r❡❧❛①❛t✐♦♥ ❛❧❧♦✇s t❤❡ ♣r♦♣❡rt② t♦ ❤♦❧❞ ❢♦r ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t♦❢ ■❊❚s✳

❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t❤❛t ❘❛t♥❡r ❉❈ ❤❛s ❢✉❧❧ ♠❡❛s✉r❡ ❢♦r ❛ s✉✐t❛❜❧❡ ❝❤♦✐❝❡ ♦❢ t❤❡♣❛r❛♠❡t❡rs✳

Pr♦♣♦s✐t✐♦♥ ✸✳✻ ✭❢✉❧❧ ♠❡❛s✉r❡ ♦❢ ❘❛t♥❡r ❉❈✮✳ ❋♦r ❛♥② τ ∈ (1, 16/15)✱ ξ ∈ (11/12, 1) ❛♥❞ η > 12 ✱ t❤❡

s❡t RDC (τ, ξ, η) ❤❛s ❢✉❧❧ ♠❡❛s✉r❡✱ s♦ ✐♥ ♣❛rt✐❝✉❧❛r ❢♦r ❡❛❝❤ ✐rr❡❞✉❝✐❜❧❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t✉♠ π ❛♥❞ ❢♦r▲❡❜❡s❣✉❡ ❛✳❡✳ λ ∈ ∆d✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ■❊❚ T = (λ, π) ❜❡❧♦♥❣s t♦ RDC (τ, ξ, η) ❛♥❞ ❤❡♥❝❡ s❛t✐s✜❡s t❤❡❘❛t♥❡r ❉❈✳

❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s r❡s✉❧t ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳✹✳ ■♥ t❤❡ ♥❡①t ❙❡❝t✐♦♥ ✸✳✸ ✇❡ ✐♥tr♦❞✉❝❡ s♦♠❡t♦♦❧s ♥❡❡❞❡❞ ✐♥ t❤❡ ♣r♦♦❢✳

✷✵ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

✸✳✸✳ ◗✉❛s✐✲❇❡r♥♦✉❧❧✐ ♣r♦♣❡rt②✳ ❚❤❡ ❜♦✉♥❞❡❞ ❞✐st♦rs✐♦♥ ♣r♦♣❡rt② ♦❢ ♣r❡✲❝♦♠♣❛❝t ❛❝❝❡❧❡r❛t✐♦♥s ♦❢t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♠❛♣ R ✭s❡❡ ❘❡♠❛r❦ ✷✳✶✸✮ ❣✉❛r❛♥t❡❡s ❛ q✉❛s✐✲❇❡r♥♦✉❧❧✐ ❦✐♥❞ ♦❢ ♣r♦♣❡rt② ✭s❡❡ ❛❧s♦❈♦r♦❧❧❛r② ✶✳✷ ✐♥ ❬✶✽❪ ♦r Pr♦♣♦s✐t✐♦♥ ✸ ✐♥ ❬✺❪✮✳

▲❡♠♠❛ ✸✳✼ ✭◗❇✲♣r♦♣❡rt②✮✳ ❋♦r ❡✈❡r② d0 > 0 t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t ν = ν(d0) > 1 s✉❝❤ t❤❛t ❢♦r❛♥② t✇♦ ♣♦s✐t✐✈❡ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝ts E,F ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♥❝❛t❡♥❛t❡❞ ✭✐✳❡✳ s✉❝❤ t❤❛t EF ✐s ❛❧s♦ ❛❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t✮ ❛♥❞ s✉❝❤ t❤❛t νcol(E) ≤ d0 ❛♥❞ t❤❡ ♣r♦❥❡❝t✐✈❡ ❞✐❛♠❡t❡rs diamH(E), diamH(F )❛r❡ ❜♦✉♥❞❡❞ ❜② d0✱ ✇❡ ❤❛✈❡

1

νµ(∆E)µ(∆F ) < µ(∆EF ) < ν µ(∆E)µ(∆F ).

Pr♦♦❢✳ ❘❡♠❛r❦ ✜rst t❤❛t ∆E ✐s ❜② ❞❡✜♥✐t✐♦♥ t❤❡ ✐♠❛❣❡ ♦❢ ∆d ✉♥❞❡r t❤❡ ♠❛♣ ψE ❣✐✈❡♥ ❜② ψE(λ) =Eλ/|Eλ|✳ ❚❤✉s✱ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡ ❢♦r♠✉❧❛ ✭r❡❝❛❧❧✐♥❣ t❤❛t ✇❡ ❞❡♥♦t❡ ❜② LebX t❤❡ ♠❡❛s✉r❡✇❤✐❝❤ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ R

d t♦ t❤❡ s✐♠♣❧❡① ∆d ❢♦r ❡❛❝❤ ♦❢ t❤❡❝♦♣✐❡s ∆π = ∆d×{π}✮ ✇❡ ❤❛✈❡ t❤❛t LebX(∆E) =

∫∆dJE(λ) dλ✱ ✇❤❡r❡ JE ❞❡♥♦t❡s t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ t❤❡

♠❛♣ ψE ✳ ❙✐♠✐❧❛r❧②✱ s✐♥❝❡ ∆EF ✐s t❤❡ ✐♠❛❣❡ ✉♥❞❡r ψE ♦❢ ∆F ✱ LebX(∆EF ) =∫∆F

JE(λ) dλ✳ ❚❤✉s✱ ❢r♦♠

▲❡♠♠❛ ✷✳✶✸✱ ✇❡ ❤❛✈❡ t❤❛t

1

νcol(E)dLebX(∆E)

LebX(∆d)<LebX(∆EF )

LebX(∆F )< νcol(E)d

LebX(∆E)

LebX(∆d).

❚❤❡ ❝❧❛✐♠ ♦❢ t❤❡ ❧❡♠♠❛ ❤❡♥❝❡ ❢♦❧❧♦✇s ❜② r❡♠❛r❦✐♥❣ t❤❛t µ ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ LebX✇✐t❤ ❞❡♥s✐t② ✇❤✐❝❤ ✐s ❜♦✉♥❞❡❞ ♦♥ ❝♦♠♣❛❝t s❡ts✳ ■♥❞❡❡❞✱ s✐♥❝❡ ∆E ✱ ∆F ❛♥❞ ∆EF ✭✇❤✐❝❤ ✐s ❝♦♥t❛✐♥❡❞✐♥ ∆E✮ ❛r❡ ♣r❡✲❝♦♠♣❛❝t ❛♥❞ ♦❢ ❞✐❛♠❡t❡r ❜♦✉♥❞❡❞ ❜② d0✱ µ(E)✱ µ(F ) ❛♥❞ µ(EF ) ❛r❡ ❝♦♠♣❛r❛❜❧❡ t♦LebX(E)✱ LebX(F ) ❛♥❞ LebX(EF ) r❡s♣❡❝t✐✈❡❧② ✇✐t❤ ❝♦♥st❛♥ts ✇❤✐❝❤ ❞❡♣❡♥❞ ♦♥ d0 ♦♥❧②✳ �

❚❤❡ t❡❝❤♥✐❝❛❧ r❡s✉❧ts t❤❛t ✇❡ ♥❡❡❞ ✐♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❛t ■❊❚s ✇✐t❤ t❤❡ ❘❛t♥❡r ❉❈ ❤❛✈❡ ❢✉❧❧ ♠❡❛s✉r❡❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ▲❡♠♠❛ ❛♥❞ ❝♦♥s❡q✉❡♥t ❈♦r♦❧❧❛r②✱ ✇❤✐❝❤ ❛r❡ ❜♦t❤ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❛❜♦✈❡ q✉❛s✐✲❇❡r♥♦✉❧❧✐ ♣r♦♣❡rt②✳ ❚❤❡② s❤♦✇ t❤❛t ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❡✈❡♥ts ❞❡s❝r✐❜❡❞ ❜② ♣r❡s❝r✐❜✐♥❣ ♠❛tr✐❝❡s ♦❢❛♥ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥ ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✐❢ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ✐s ♣r❡✲❝♦♠♣❛❝t✳

▲❡♠♠❛ ✸✳✽✳ ▲❡t A = AY ❜❡ ❛ ♣r❡✲❝♦♠♣❛❝t ❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥

R✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c = c(Y ) > 1 s✉❝❤ t❤❛t ❢♦r ❛♥② ✐♥t❡❣❡rs 0 ≤ l < m < n ❛♥❞ ❛♥② ♠❛tr✐❝❡sC0, C1 ∈ SL(d,Z) ✇❡ ❤❛✈❡ t❤❛t

✭✸✳✺✮ µ(T ∈ Y : AlAl+1 · · ·Am−1(T ) = C0, AmAm+1 · · ·An(T ) = C1

)

≤ c · µ(T ∈ Y : AlAl+1 · · ·Am−1(T ) = C0

)· µ(T ∈ Y : AmAm+1 · · ·An(T ) = C1

).

❈♦r♦❧❧❛r② ✸✳✾✳ ❋♦r ❛♥② ♣r❡✲❝♦♠♣❛❝t ❛❝❝❡❧❡r❛t✐♦♥ A = AY ❛♥❞ ❛♥② N ∈ N t❤❡r❡ ❡①✐sts c = c(Y , N)s✉❝❤ t❤❛t ❢♦r ❛♥② ❝❤♦✐❝❡ ♦❢ ✐♥t❡❣❡rs l1 < l2 < · · · < lN ❛♥❞ i1, . . . , iN ✇❡ ❤❛✈❡ t❤❛t

✭✸✳✻✮ µ(T ∈ Y : ‖Al1(T )‖ = i1, ‖Al2(T )‖ = i2, . . . , ‖AlN (T )‖ = iN

)

≤ c · µ(T ∈ Y : ‖Al1(T )‖ = i1

)· · · µ

(T ∈ Y : ‖AlN (T )‖ = iN

).

▲❡t ✉s ✜rst ❣✐✈❡ t❤❡ ♣r♦♦❢ ♦❢ ❈♦r♦❧❧❛r② ✸✳✾✱ t❤❡♥ t❤❡ ♦♥❡ ♦❢ ▲❡♠♠❛ ✸✳✽✳

Pr♦♦❢ ♦❢ ❈♦r♦❧❧❛r② ✸✳✾✳ ▲❡t C1, . . . , CN ❜❡ ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s ❛♥❞ D1, . . . , DN−1 ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✲

✉❝ts s✉❝❤ t❤❛t ❢♦r s♦♠❡ T ∈ Y ✇❡ ❤❛✈❡

Ali(T ) = Ci ❢♦r 1 ≤ i ≤ N, Di = Ali+1 · · ·Ali+1−1 ❢♦r 1 ≤ i ≤ N − 1,

❙♦ t❤❛t ✐♥ ♣❛rt✐❝✉❧❛r Al1Al1+1 · · ·AlN (T ) = C1D1C2D2 · · ·CN−1DN−1CN ✳ ❇② ❛♣♣❧②✐♥❣ ▲❡♠♠❛ ✸✳✽2N − 1 t✐♠❡s ✐♥ ♦r❞❡r t♦ s♣❧✐t ✉♣ t❤❡ ♣r♦❞✉❝t ✐♥t♦ ❛ ♣r♦❞✉❝t ♦❢ ♠❛tr✐❝❡s Ci ❛♥❞ Di ✭♠♦r❡ ♣r❡❝✐s❡❧②

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✷✶

❛♣♣❧②✐♥❣ ✐t t♦ Ci ❛♥❞ Di · · ·CN ♦r Di ❛♥❞ Ci+1 · · ·CN ❢♦r i = 1, . . . , N − 1✮ ✇❡ ❤❛✈❡ t❤❛t

µ(T ∈ Y : Al1Al1+1 · · ·AlN (T ) = C1D1C2D2 · · ·CN−1DN−1CN

)

≤ c(Y )2N−1 ·ΠNi=1µ(T ∈ Y : Ali(T ) = Ci

)ΠN−1i=1 µ

(T ∈ Y : Ali+1 · · ·Ali+1−1(T ) = Di

).

❋♦r ❜r❡✈✐t② ❧❡t ✉s ❞❡♥♦t❡ ❜② Ci(Di) t❤❡ ❡✈❡♥t {T ∈ Y : Ali+1 · · ·Ali+1−1(T ) = Di}✳ ❚❤❡♥✱ s✉♠♠✐♥❣ ♦✈❡r❛❧❧ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ♦❢ t❤❡ ♠❛tr✐❝❡s D1, . . . , DN−1 ❛♥❞ ✉s✐♥❣ t❤❛t t❤❡ ❡✈❡♥ts Ci(Di)) ❢♦r 1 ≤ i ≤ N − 1❛r❡ ❞✐s❥♦✐♥t✱ ✇❡ ❣❡t t❤❛t

∑

D1,...,DN−1

ΠN−1i=1 µ(Ci(Di)) ≤ ΠN−1

i=1

∑

D1,...,DN−1

µ(Ci(Di))

= ΠN−1

i=1 µ(∪N−1i=1 Ci(Di)

)≤ µ(Y )N−1

❙✉♠♠✐♥❣ ✉♣ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ♦❢ ♠❛tr✐❝❡s C1, . . . , CN ❛♥❞ D1, . . . , DN−1 ❛s ❛❜♦✈❡ ❛♥❞ s✉❝❤ t❤❛t

‖Cj‖ = ij ❢♦r j = 1, . . . , N ✇❡ ❣❡t ✭✸✳✻✮ ❢♦r c = c(Y , N) + c(Y )2N−1µ(T )N−1✳ �

❚❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✸✳✽ ✐s ❜❛s❡❞ ♦♥ t❤❡ r❡♠❛r❦ t❤❛t ✐♥ ❛ ♣r❡✲❝♦♠♣❛❝t ❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥

A = AY ✱ ❡✈❡r② r❡t✉r♥ ♦❢ t❤❡ ♦r❜✐t ♦❢ T t♦ Y ✱ ✐✳❡✳ ❡✈❡r② l s✉❝❤ t❤❛t RlY(T ) ∈ Y ✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦

✈✐s✐ts ♦❢ RlY(T ) t♦ ❛ ❝②❧✐♥❞❡r (ΘC ×∆D)

(1) ✇❤❡r❡ C,D ❛r❡ ♣♦s✐t✐✈❡ ♠❛tr✐❝❡s ✇✐t❤ ✉♥✐❢♦r♠❡❧② ❜♦✉♥❞❡❞

❍✐❧❜❡rt ❞✐❛♠❡t❡r ❛♥❞ νcol✳ ❊q✉✐✈❛❧❡♥t❧②✱ t❤✐s ♠❡❛♥s t❤❛t ♦♥❡ s❡❡s t❤❡ ❜❧♦❝❦ CD ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❝♦❝②❝❧❡♣r♦❞✉❝ts ❝❡♥t❡r❡❞ ❛t t✐♠❡ l✱ ✐✳❡✳ t❤❡ ❝♦❝②❝❧❡ ♠❛tr✐❝❡s Al+L · · ·Al−2Al−1 ❡♥❞ ✇✐t❤ C ❛♥❞ AlAl+1, . . . Al+Lst❛rt ✇✐t❤ D ❢♦r s♦♠❡ L✻✳ ❚❤✐s ❤❡♥❝❡ ❛❧❧♦✇s t♦ ✉s❡ t❤❡ ◗❇✲♣r♦♣❡rt②✳

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✸✳✽✳ ❘❡♠❛r❦ ✜rst t❤❛t ❜② ❞❡✜♥✐t✐♦♥

AlAl+1 · · ·Am−1(T ) = A0A1 · · ·Am−l−1(RlY(π, λ, τ)),

s♦ t❤❛t✱ s✐♥❝❡ t❤❡ ♠❡❛s✉r❡ µ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r R ❛♥❞ ❤❡♥❝❡ ✐ts r❡str✐❝t✐♦♥ t♦ Y ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡P♦✐♥❝❛ré ♠❛♣ RY ✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❡ ▲❡♠♠❛ st❛t❡♠❡♥t ❢♦r l = 0 ✭✇❤❡r❡ 0 < m < n ♣❧❛② t❤❡

r♦❧❡ ♦❢ t❤❡ ❢♦r♠❡r 0 < m− l < n− l✮✳ ❙✐♥❝❡ A = AY ✐s ❛ ❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ R✱ Y ✐s t❤❡ ✉♥✐♦♥ ♦❢

✜♥✐t❡❧② ♠❛♥② ❝②❧✐♥❞❡rs ❢♦r R✱ t❤❛t ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② (ΘEk×∆Fk)(1) ❢♦r 1 ≤ k ≤ N ✳ ❋✉rt❤❡r❡♠♦r❡ s✐♥❝❡

A ✐s ♣r❡❝♦♠♣❛❝t ❛❧❧ ♠❛tr✐❝❡s Ei ❛♥❞ Fi ❛r❡ ♣♦s✐t✐✈❡ ❛♥❞ ❤❡♥❝❡ ❤❛✈❡ ✜♥✐t❡ ❍✐❧❜❡rt ❞✐❛♠❡t❡r ❜② ❘❡♠❛r❦✷✳✻✳ ▲❡t dY ❜❡ t❤❡ ♠❛①✐♠✉♠ ♦❢ t❤❡ ♣r♦❥❡❝t✐✈❡ ❞✐❛♠❡t❡rs diamH(Ei), diamH(Fi) ❢♦r 1 ≤ i ≤ N ❛♥❞ ♦❢νcol(Ei) ❢♦r 1 ≤ i ≤ N ✳ ▲❡t νY + ν(dY ) ❜❡ t❤❡ ❝♦♥st❛♥t ❣✐✈❡♥ ❜② ▲❡♠♠❛ ✸✳✼✳ ▲❡t n0 ❜❡ t❤❡ ♠❛①✐♠❛❧♥✉♠❜❡r ♦❢ ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s ♣r♦❞✉❝❡❞ t♦ ♦❜t❛✐♥ ❛♥② ♦❢ t❤❡ ♠❛tr✐❝❡s Ei ♦r Fi ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢t❤❡ ❝②❧✐♥❞❡rs✳ ◆♦t✐❝❡ t❤❛t ✐t ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈❡ t❤❡ st❛t❡♠❡♥t ♦❢ t❤❡ ▲❡♠♠❛ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t❡✐t❤❡r m ≥ n0 ♦r n −m ≥ n0✱ s✐♥❝❡ t❤❡ ♣♦ss✐❜❧❡ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝ts ♦❢ n0 ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s❛r❡ ✜♥✐t❡❧② ♠❛♥② ❛♥❞ ❤❡♥❝❡ t❤❡ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦ss✐❜✐❧✐t✐❡s ✇✐t❤ m ≤ n0 ❛♥❞ n ≤ m+ n0 ≤ 2n0 ♦♥❧②❝❤❛♥❣❡ t❤❡ ❝♦♥st❛♥t c✳

❚❤❡ ❝♦❝②❝❧❡ A = AY ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✜rst r❡t✉r♥ ♠❛♣ RY t♦ Y ✐s ❧♦❝❛❧❧② ❝♦♥st❛♥t ❛♥❞✱ ✇❤❡♥

r❡str✐❝t❡❞ t♦ ♦♥❡ ♦❢ t❤❡ ❝②❧✐♥❞❡rs (ΘEk ×∆Fk)(1) ✐♥ Y ✱ t❤❡ s❡t ✇❤❡r❡ ✐t t❛❦❡s ❛s ❛ ✈❛❧✉❡ ❛ ✜①❡❞ ❣✐✈❡♥

❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t C ♦❢ p ♠❛tr✐❝❡s ❛♥❞✱ ❛❢t❡r p ✐t❡r❛t✐♦♥s ♦❢ RY ♦♥❡ ❧❡♥❞s t♦ (ΘEl ×∆Fl)(1)✱ ✐✳❡✳ t❤❡

s❡t

C = C(C, k, l) +{T ∈ (ΘEk ×∆Fk)

(1) s.t. A(T ) = C, RpY(T ) ∈ (ΘEl ×∆Fl)

(1)}

✐s ❛ ❘❛✉③②✲❱❡❡❝❤ ❝②❧✐♥❞❡r ♦❢ t❤❡ ❢♦r♠ (ΘD × ∆CFl)(1)✱ ✇❤❡r❡ ΘD ⊂ ΘEk ❛♥❞ ∆CFk ⊂ ∆Fk ✭✇❤✐❝❤

❡①♣❧✐❝✐t❧② ♠❡❛♥s t❤❛t t❤❡ ♣r♦❞✉❝t CFl st❛rts ✇✐t❤ Fk✱ ✐✳❡✳ ✐t ❤❛s t❤❡ ❢♦r♠ FkC′ ❢♦r s♦♠❡ ♥♦♥✲♥❡❣❛t✐✈❡

C ′✱ ❛♥❞ t❤❡ ♣r♦❞✉❝t D ❡♥❞s ✇✐t❤ Ek ✐✳❡✳ ✐t ❤❛s t❤❡ ❢♦r♠ D′Ek ❢♦r s♦♠❡ ♥♦♥ ♥❡❣❛t✐✈❡ D′✮ ❛♥❞✱ s✐♥❝❡❜② ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝♦❝②❝❧❡ ❛♥❞ ❝②❧✐♥❞❡rs ✭s❡❡ ✐♥ ♣❛rt✐❝✉❧❛r ✭✷✳✼✮✮✱ r❡❝❛❧❧✐♥❣ t❤❛t p ✐s t❤❡ ♥✉♠❜❡r ♦❢❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s ♣r♦❞✉❝❡❞ t♦ ❣❡t C✱ ✇❡ ❤❛✈❡ t❤❛t

✻❖♥❡ ❝❛♥ ♠❛❦❡ s♣❡❝✐❛❧ ❝❤♦✐❝❡s ♦❢ ❝②❧✐♥❞r✐❝❛❧ ❛❝❝❡❧❡r❛t✐♦♥s ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❆✈✐❧❛✲●♦✉❡③❡❧✲❨♦❝❝♦③ ❬✸❪ ✇❤✐❝❤ ❣✉❛r❛♥t❡❡t❤❛t Al ✭r❡s♣✳ Al−1✮ ✐s s✉✣❝✐❡♥t❧② ❧♦♥❣ s♦ t❤❛t ✐t ❤❛s t♦ st❛rt ✭r❡s♣✳ ❡♥❞✮ ✇✐t❤ s♦♠❡ s♣❡❝✐✜❝ ♠❛tr✐❝❡s D ✭r❡s♣✳ C✮✳ ■♥❣❡♥❡r❛❧✱ ✐t ♠✐❣❤t ❤❛♣♣❡♥ t❤♦✉❣❤ t❤❛t t♦ s❡❡ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ C ♦r D ♦♥❡ ♥❡❡❞s t♦ ❝♦♥s✐❞❡r s❡✈❡r❛❧ s✉❝❝❡ss✐✈❡ st❡♣s✱ ✇❤✐❝❤❝♦♠♣❧✐❝❛t❡s t❤❡ ✇r✐t✐♥❣ ♦❢ t❤❡ ♣r♦♦❢✮

✷✷ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

Rp

Y

((ΘD ×∆CFl)

(1))= (ΘDC ×∆Fl)

(1) ⊂ (ΘEl ×∆Fl)(1),

t❤❡ ♠❛tr✐① DC ❡♥❞s ✇✐t❤ El✿ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ❡✐t❤❡r D = Ek ❛♥❞ ❤❡♥❝❡ DC = EkC = C ′′El ❢♦r s♦♠❡♥♦♥ ♥❡❣❛t✐✈❡ C ′′ s♦ t❤❛t ΘDC ⊂ ΘEl ✭✇❤✐❝❤ ❤❛♣♣❡♥s ✐❢ t❤❡ ♠❛tr✐① C ✐s s✉✣❝❡♥t❧② ❧❛r❣❡ s♦ t❤❛t EkC✐s ❛ ❧♦♥❣❡r ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t t❤❛♥ El✮✱ ♦r ♦t❤❡r✇✐s❡ ΘD ✐s ❛ str✐❝t s✉❜s❡t ♦❢ ΘEk ❝❤♦s❡♥ s♦ t❤❛tDC = El ❛♥❞ ❤❡♥❝❡ ΘDC = ΘEl ✳

▲❡t C0, C1 ❜❡ ❛♥② t✇♦ ♠❛tr✐❝❡s ✐♥ SL(d,Z)✳ ❘❡♠❛r❦ t❤❛t ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t C0, C1 ❛r❡ ❘❛✉③②✲

❱❡❡❝❤ ♣r♦❞✉❝ts t❤❛t ❝❛♥ ❜❡ ❝♦♥❝❛t❡♥❛t❡❞✱ ✐✳❡✳ t❤❛t C0 = A0A1 · · ·Am−1(T ) ❛♥❞ C1 = Am · · ·An−1(T )

❢♦r s♦♠❡ ✭❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ s❡t ♦❢✮ T = (τ, λ, π) ∈ Y ✮✱ s✐♥❝❡ ♦t❤❡r✇✐s❡ ✐♥ t❤❡ st❛t❡♠❡♥t ❡✐t❤❡r t❤❡ ❘❍❙♦r ❜♦t❤ s✐❞❡s ❛r❡ ③❡r♦ ❛♥❞ t❤❡ st❛t❡♠❡♥t ✐s tr✐✈✐❛❧❧② tr✉❡✳ ▲❡t C0 ❛♥❞ C1 ❜❡ t❤❡ ❝②❧✐♥❞❡rs ❛s ❞❡s❝r✐❜❡❞❛❜♦✈❡ ♦♥ ✇❤✐❝❤ t❤❡ ❝♦❝②❝❧❡ A ✐s ❧♦❝❛❧❧② ❡q✉❛❧ t♦ C0 ❛♥❞ C1 r❡s♣❡❝t✐✈❡❧② ❛♥❞ s✉❝❤ t❤❛t Rm

YC0 ∩ C1 6= ∅✳

■♥ ✈✐rt✉❡ ♦❢ t❤❡ r❡♠❛r❦ ❛❜♦✈❡✱ ✇❡ ✇r✐t❡ ❢♦r s♦♠❡ 1 ≤ k0, k1, k2 ≤ N ✱

Ci = (ΘDi ×∆CiFki)(1) ⊂ (ΘEki

×∆Fki)(1) ❢♦r i = 0, 1 ✇❤❡r❡

✭✸✳✼✮ ❡✐t❤❡r D1 = Ek1 , ♦r D1 ❡♥❞s ✇✐t❤ Ek1 ❛♥❞ D1C1 = Ek2 .

❙✐♠✐❧❛r❧②✱ s✐♥❝❡ C0 ✐s t❤❡ ♣r♦❞✉❝t ♦❢ m ❘❛✉③②✲❱❡❡❝❤ ♠❛tr✐❝❡s✱ ♦♥❡ ❝❛♥ s❡❡ t❤❛t

C0 ∩ R−m

YC1 = (ΘD ×∆C0C1F )

(1),

✇❤❡r❡ ✭❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ ❜❡t✇❡❡♥ C1Fk2 ♦r Fk1 ✐s ❛ ❧♦♥❣❡r ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t✮

✭✸✳✽✮ ❡✐t❤❡r F = Fk2 ❛♥❞ C1F st❛rts ✇✐t❤ Fk1 , ♦r C1F = Fk1 ❛♥❞ F st❛rts ✇✐t❤ Fk2 ,

❛♥❞ D ✐s s✉❝❤ t❤❛t

❡✐t❤❡r D = Ek0 , ❛♥❞ DC0 ❛♥❞ DC0C1 ❡♥❞ ✇✐t❤ Ek1 ❛♥❞ Ek2 r❡s♣❡❝t✐✈❡❧②,

♦r DC0 = Ek1 , ❛♥❞ D ❛♥❞ DC0C1 ❡♥❞ ✇✐t❤ Ek0 ❛♥❞ Ek2 r❡s♣❡❝t✐✈❡❧②;✭✸✳✾✮

✭t❤❡ ❧❛st ♣♦ss✐❜✐❧✐t②✱ ✐✳❡✳ t❤❛t DC0C1 = Ek2 ✐s ❡①❝❧✉❞❡❞ s✐♥❝❡ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❡✐t❤❡r m ≥ n0 ♦rn−m ≥ n0 ❡♥s✉r❡ t❤❛t C0C1 ✐s ❛ s✉✣❝✐❡♥t❧② ❧♦♥❣ ❘❛✉③②✲❱❡❡❝❤ ♣r♦❞✉❝t t♦ ❜❡❣✐♥ ✇✐t❤ Ek2✮✳

▲❡t ✉s ♥♦✇ ❝♦♠♣❛r❡ t❤❡ ♠❡❛s✉r❡s ♦❢ C0,C1 ❛♥❞ C0∩R−m

YC1✳ ❯s✐♥❣ t❤❛t µ ✐s R ✐♥✈❛r✐❛♥t✱ t❤❡ ❞❡✜♥✐t✐♦♥

♦❢ ❝②❧✐♥❞❡rs ✭s❡❡ ✐♥ ♣❛rt✐❝✉❧❛r ✭✷✳✼✮✮ ❛♥❞ t❤❛t µ = p∗µ✱ ✇❡ ❤❛✈❡ t❤❛t ❢♦r l = 0, 1✱ ✐❢ Dl ✐s ❛ ❘❛✉③②✲❱❡❡❝❤♣r♦❞✉❝t st❛rt✐♥❣ ❛t πl

✭✸✳✶✵✮ µ(Cl) = µ((ΘDl ×∆ClFkl+1

)(1))= µ

((Θπl ×∆DlCjFkl+1

)(1))= µ

(∆DlClFkl+1

).

❙✐♠✐❧❛r❧②✱ µ(C0 ∩ R−m

YC1) = µ(∆DC0C1F )✳ ❲❡ ❝❛♥ ♥♦✇ ❣❡t t❤❛t

µ(C0 ∩ R−m

YC1) ≤ νY µ(∆DC0)µ(∆C1F ).

❜② ❛♣♣❧②✐♥❣ t❤❡ ✉♣♣❡r ✐♥❡q✉❛❧✐t② ✐♥ ▲❡♠♠❛ ✸✳✼ ❛♥❞ r❡❝❛❧❧✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ νY ✿ t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ t❤❡▲❡♠♠❛ ❤♦❧❞s s✐♥❝❡ νcol(DC0) ≤ νcol(Ek1) ❜② ✭✸✳✾✮ ❛♥❞ ✭✷✳✶✷✮✱ diamH(DC0) ≤ max{diamH(Ek0), diamH(Ek1)}❜② ✭✸✳✾✮ ❛♥❞ ❘❡♠❛r❦ ✷✳✼ ❛♥❞ diamH(C1F ) ≤ diamH(Fk1) ❜② ✭✸✳✽✮ ❛♥❞ ❘❡♠❛r❦ ✷✳✼✳ ◆♦✇✱ ❞✐✈✐❞✐♥❣ ❛♥❞♠✉❧t✐♣❧②✐♥❣ ❜② µ(∆Fk1

) ❛♥❞ µ(∆D1) ❛♥❞ ❜② r❡♠❛r❦✐♥❣ t❤❛t ❜② ✭✸✳✼✮ ❡✐t❤❡r ∆D1 = ∆Ek1♦r ♦t❤❡r✇✐s❡

∆D1 ⊃ ∆D1C1 = ∆Ek2❛♥❞ ❤❡♥❝❡ ✐♥ ❜♦t❤ ❝❛s❡s µ(D1) ≥ inf1≤k≤N µ(∆Ek)✱ ✇❡ ❣❡t t❤❛t

✭✸✳✶✶✮ µ(C0 ∩ R−1

YC1) ≤ νY

µ(∆DC0)µ(∆Fk1)µ(∆D1)µ(∆C1F )

inf1≤i≤N µ(∆Fi) inf1≤i≤N µ(∆Ei).

❆♣♣❧②✐♥❣ ♥♦✇ t❤❡ ❧♦✇❡r ✐♥❡q✉❛❧✐t② ✐♥ ▲❡♠♠❛ ✸✳✼ ✭✇❤✐❝❤ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t❤❛♥❦s t♦ ✭✸✳✼✮✱ ✭✸✳✽✮ ❛♥❞ t❤❡❞❡✜♥✐t✐♦♥ ♦❢ νY ✱ ✇❤✐❝❤ ❣✐✈❡ t❤❛t νcol(D1) ≤ νcol(Ek1) ❜② ✭✷✳✶✷✮ ❛♥❞ ❛❧❧♦✇ t♦ ❡st✐♠❛t❡ ❞✐❛♠❡t❡rs ✉s✐♥❣❘❡♠❛r❦ ✷✳✼✮✱ ❛♥❞ t❤❡♥ r❡♠❛r❦✐♥❣ t❤❛t ✭✸✳✽✮ ✐♠♣❧✐❡s t❤❛t ∆D1C1F ⊂ ∆D1C1Fk2

❛♥❞ ✉s✐♥❣ ✭✸✳✶✵✮ ❢♦rl = 1✱ ✇❡ ❤❛✈❡ t❤❛t

✭✸✳✶✷✮ µ(∆D1)µ(∆C1F ) ≤ νY µ(∆D1C1F ) ≤ νY µ(∆D1C1Fk2) = νY µ(C1).

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✷✸

❙✐♠✐❧❛r❧②✱ ❛❣❛✐♥ ❜② t❤❡ ❧♦✇❡r ✐♥❡q✉❛❧✐t② ✐♥ ▲❡♠♠❛ ✸✳✼ ✭✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ t❤✐s t✐♠❡ t❤❛♥❦s t♦ ✭✸✳✾✮✱✭✷✳✶✷✮ ❛♥❞ ❘❡♠❛r❦ ✷✳✼✱ ✇❤✐❝❤ ✐♥ ♣❛rt✐❝✉❧❛r ②✐❡❧❞ νcol(DC0) ≤ νcol(Ek1)✮✱ r❡❛s♦♥✐♥❣ ❛s ✐♥ ✭✸✳✶✵✮ ❛♥❞t❤❡♥ r❡♠❛r❦✐♥❣ t❤❛t ΘD ⊂ ΘEk0

❜② ✭✸✳✾✮✱ ✇❡ ❛❧s♦ ❤❛✈❡ t❤❛t

✭✸✳✶✸✮ µ(∆DC0)µ(∆Fk1) ≤ νY µ(∆DC0Fk1

) = νY µ((ΘD ×∆C0Fk1

)(1))≤ νY µ(C0).

❈♦♠❜✐♥✐♥❣ ✭✸✳✶✶✮✱ ✭✸✳✶✷✮ ❛♥❞ ✭✸✳✶✸✮✱ ✇❡ ✜♥❛❧❧② ❣❡t

✭✸✳✶✹✮ µ(C0 ∩ R−m

YC1) ≤ c µ(C0) µ(C1), ✇❤❡r❡ c = c(Y ) + ν3

Y/( inf

1≤i≤Nµ(∆Fi) inf

1≤i≤Nµ(∆Ei)).

❖♥❡ ❝❛♥ ♥♦✇ ❝♦♥❝❧✉❞❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛ ❜② s✉♠♠✐♥❣ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ♦❢ s②♠♣❧❡①❡sC0,C1 ❛s ❛❜♦✈❡✱ ♥❛♠❡❧② ❜② s✉♠♠✐♥❣ ♦✈❡r ❛❧❧ ❝❤♦✐❝❡s ♦❢ ❝②❧✐♥❞❡rs C(C1, k1, l1) ❛♥❞ C(C2, k2, l2) ❢♦r1 ≤ k1, k2, l1, l2 ≤ N ✳ �

✸✳✹✳ ❋✉❧❧ ♠❡❛s✉r❡ ♦❢ t❤❡ ❘❛t♥❡r ❉❈✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ♣r♦✈❡ Pr♦♣♦s✐t✐♦♥ ✸✳✻✱ ❜② s❤♦✇✐♥❣ t❤❛tt❤❡ ❘❛t♥❡r ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ❢♦r ❛ s✉✐t❛❜❧❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs τ, ξ ❛♥❞ η ✐s s❛t✐s✜❡❞ ❜② ❛ ❢✉❧❧♠❡❛s✉r❡ s❡t ♦❢ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✳

Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✻✳ ❙❡t δ + 1− τ−1 ❛♥❞ ❝♦♥s✐❞❡r t❤❡ s❡t Y + Yδ ❣✐✈❡♥ ❜② ❚❤❡♦r❡♠ ✷✳✶✹✱ s♦ t❤❛tt❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛❝❝❡❧❡r❛t✐♦♥ A + AY ❤❛s t❤❡ ❡①♣♦♥❡♥t✐❛❧ t❛✐❧s ♣r♦♣❡rt② ✭s❡❡ ❚❤❡♦r❡♠ ✷✳✶✹✮✳ ❙✐♥❝❡❜② t❤❡ ❝❤♦✐❝❡ ♦❢ τ ✇❡ ✐♥ ♣❛rt✐❝✉❧❛r ❤❛✈❡ t❤❛t 1 < τ < 2✱ ❜② Pr♦♣♦s✐t✐♦♥ ✸✳✸ ✭s❡❡ ❛❧s♦ ❘❡♠❛r❦ ✸✳✹✮✱ t❤❡r❡

❡①✐sts ❛ s✉❜s❡t Y ′ ⊂ Y ✇✐t❤ µ(Y ′) = µ(Y ) s✉❝❤ t❤❛t ❢♦r ❛♥② T ∈ Y ′ t❤❡ s❡q✉❡♥❝❡ (nl)l ♦❢ r❡t✉r♥s ♦❢ T

t♦ Y ′ s❛t✐s✜❡s t❤❡ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❉❡✜♥✐t✐♦♥ ✸✳✶ ♦❢ t❤❡ ▼✐①✐♥❣ ❉❈ ♣r♦♣❡rt② ✇✐t❤ ✐♥t❡❣r❛❜✐❧✐t② ♣♦✇❡rτ ❢♦r s♦♠❡ ✜①❡❞ ℓ ∈ N ❛♥❞ η > 1✳ ❘❡❝❛❧❧ t❤❛t L = ℓ(1 +

[logd(2ν

2)]) ✭s❡❡ ✭✸✳✹✮✮✳ ❋✐① k ∈ N✳

❈❧❛✐♠✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c = c(L) > 0 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② R > 0 ❛♥❞ ❡✈❡r② 0 ≤ J ≤ L ✇❡❤❛✈❡

✭✸✳✶✺✮ µ(T ∈ Y : ‖Ak(T )‖ . . . ‖Ak+J(T )‖ > R) <c

R1−δ.

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❈❧❛✐♠ ❣♦❡s ❜② ✐♥❞✉❝t✐♦♥ ♦♥ J ✳ ❋♦r J = 0 ❜② t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✭✷✳✶✸✮ ♦❢ A

✭❣✐✈❡♥ ❜② ❚❤❡♦r❡♠ ✷✳✶✹✮ ❛♥❞ ❜② ✐♥✈❛r✐❛♥❝❡ ♦❢ µ ✉♥❞❡r R✱ ❢♦r ❛♥② R ∈ N ✇❡ ❤❛✈❡ t❤❛t

R1−δ µ(T ∈ Y s.t. ‖Ak(T )‖ > R) ≤∑

i>R

i1−δ µ(T ∈ Y s.t. ‖A(T )‖ = i) ≤

∫

Yδ

‖A‖1−δ dµ + c0.

❆ss✉♠❡ t❤❛t t❤❡ ❈❧❛✐♠ ❤♦❧❞s ❢♦r J < L✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t ✐t ❤♦❧❞s ❢♦r J + 1✳ ❇② s✉♠♠✐♥❣ t❤❡ ◗❇♣r♦♣❡rt② ❢♦r t❤❡ ❝♦❝②❝❧❡ A ♣r♦✈❡❞ ✐♥ ▲❡♠♠❛ ✸✳✽ ♦✈❡r t❤❡ s❡t ♦❢ j s✉❝❤ t❤❛t j > R/i✱ ✇❡ ❤❛✈❡ t❤❛t

µ(T ∈ Y : ‖Ak(T )‖ . . . ‖Ak+J+1(T )‖ > R) ≤

R∑

i=1

µ(T ∈ Y : ‖Ak+J+1(T )‖ = i ❛♥❞ ‖Ak(T )‖ . . . ‖Ak+J(T )‖ > R/i) ≤

cY

R∑

i=1

µ(T ∈ Y : ‖Ak+J+1(T )‖ = i) µ(T ∈ Y : ‖Ak(T )‖ . . . ‖Ak+J(T )‖ > R/i).

✭✸✳✶✻✮

❇② t❤❡ ✐♥❞✉❝t✐♦♥ ❛ss✉♠♣t✐♦♥ ✭✐✳❡✳ ✉s✐♥❣ t❤❡ ❈❧❛✐♠ ❢♦r J ❛♥❞ R/i✱ i = 1 . . . R✮ ✇❡ ❣❡t t❤❛t

µ(T ∈ Y : ‖Ak(T )‖ . . . ‖Ak+J(T )‖ > R/i) ≤ c

(i

R

)(1−δ)

, ❢♦r i = 1, . . . , R.

✷✹ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

❚❤❡r❡❢♦r❡✱ ❞❡♥♦t✐♥❣ ❜② Yi t❤❡ s❡t ♦❢ T = (τ, λ, π) ∈ Y s✉❝❤ t❤❛t ‖A(T )‖ = i✱ ✇❡ ❤❛✈❡ ❜② t❤❡

✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✭✷✳✶✸✮ ♦❢ A ✭❣✐✈❡♥ ❜② ❚❤❡♦r❡♠ ✷✳✶✹✮ ❛♥❞ ❜② ✐♥✈❛r✐❛♥❝❡ ♦❢ µ ✉♥❞❡r R

µ(T : ‖Ak(T )‖ . . . ‖Ak+J+1(T )‖ > R) ≤cY c

R1−δ

R∑

i=1

i1−δ µ(T : ‖Ak+J+1(T )‖ = i)

=cY c

R1−δ

R∑

i=1

∫

Yi

i1−δ ≤cY c

R1−δ

∫

Y‖A‖1−δ ≤

cY c2

R1−δ.

❚❤✐s ✜♥✐s❤❡s t❤❡ ♣r♦♦❢ ♦❢ ❈❧❛✐♠✳ ❯s✐♥❣ ✭✸✳✶✺✮ ❢♦r J = L ❛♥❞ R = kξ✱ ✇❡ ❣❡t ✐♥ ♣❛rt✐❝✉❧❛r

✭✸✳✶✼✮ µ(T ∈ Y : ‖Ak(T )‖ . . . ‖Ak+L(T )‖ > kξ) <c

kξ(1−δ).

◆♦✇ ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t ❢♦r µ✲❛❧♠♦st ❡✈❡r② T ∈ Y ✱ ✐♥ ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ❢♦r♠ [j2, (j + 1)2] ✭j s✉✛✳

❧❛r❣❡✮ t❤❡r❡ ❛r❡ ❛t ♠♦st 2L ✐♥❞❡①❡s k ∈ [j2, (j + 1)2] s✉❝❤ t❤❛t ‖Ak(T )‖ . . . ‖Ak+L(T )‖ > kξ✳ ❚❤✐s

❢♦❧❧♦✇s ❜② ❈♦r♦❧❧❛r② ✸✳✾ ❛s t❤❡ ❡✈❡♥ts ‖Ak(T )‖ . . . ‖Ak+L(T )‖ > kξ✱ ‖Al(T )‖‖Al+L(T )‖ > lξ ❛r❡ ❛❧♠♦st✐♥❞❡♣❡♥❞❡♥t ❢♦r l /∈ {k − L, . . . , k + L}✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts c = c(L) > 0 s✉❝❤ t❤❛t

µ(T ∈ Y : ‖Ak(T )‖ . . . ‖Ak+L(T )‖ > kξ, ‖Al(T )‖ . . . ‖Al+L(T )‖ > lξ)

≤ cµ(T ∈ Y : ‖Ak(T )(T )‖ . . . ‖Ak+L(T )‖ > kξ)µ(T ∈ Y : ‖Al(T )‖ . . . ‖Al+L(T )‖ > lξ).✭✸✳✶✽✮

❚♦ s❤♦✇ t❤✐s ♥♦t✐❝❡ t❤❛t ✇❡ ❝❛♥ ❞❡❝♦♠♣♦s❡ t❤❡ ▲❍❙ ❛♥❛❧♦❣♦✉s❧② t♦ ✭✸✳✶✻✮ ❛♥❞ t❤❡♥✱ s✐♥❝❡ l /∈ {k −L, . . . , k + L}✱ ✉s❡ ❈♦r♦❧❧❛r② ✸✳✾✳

❋♦r k ≥ 1✱ ❞❡♥♦t❡ ❜② Ek t❤❡ s❡t ♦❢ T ∈ Y s✉❝❤ t❤❛t ‖Ak(T )‖ . . . ‖Ak+L(T )‖ > kξ✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞✐♥

Fj = {T ∈ Y s.t. T ∈ Ek ❤♦❧❞s ❢♦r ❛t ❧❡❛st 2L+ 1 ♥✉♠❜❡rs k ✐♥ [j2, (j + 1)2]}, j ≥ 1.

❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s❡t Fj ✱ ✐t ❢♦❧❧♦✇s t❤❛t

✭✸✳✶✾✮ Fj ⊂∑

m,n∈[j2,(j+1)2],n/∈{m−L,...,m+L}

Em ∩ En.

◆♦✇ ❜② ✭✸✳✶✼✮ ❛♥❞ ✭✸✳✶✽✮✱ ✇❡ ❤❛✈❡ t❤❛t ❢♦r ❛♥② m ❛♥❞ n /∈ {m− L, . . . ,m+ L}✱

µ(Em ∩ En) ≤

(1

mn

)ξ(1−δ)≤

1

j4ξ(1−δ).

❚❤❡r❡❢♦r❡✱ ❜② ✭✸✳✶✾✮✱ ✇❡ ❤❛✈❡

µ(Fj) ≤4j2

j4ξ(1−δ)≪

1

j1+1/100❢♦r δ <

1

16❛♥❞ ξ ≥> 11/12.

❍❡♥❝❡✱∑

j≥1 µ(Fj) < +∞✳ ❚❤✉s✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❇♦r❡❧✲❈❛♥t❡❧❧✐ ▲❡♠♠❛ t❤❛t t❤❡r❡ ❡①✐sts ❛ s❡t

Y ′′ ∈ Y ✇✐t❤ µ(Y ′′) = µ(Y ) s✉❝❤ t❤❛t ❛♥② T ∈ Y ′′ ❜❡❧♦♥❣s ♦♥❧② ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t♦⋃j≥1 Fj ✳

❉❡♥♦t❡ ❜② nT ∈ N t❤❡ ❧❛st ✈✐s✐t ♦❢ T t♦⋃j≥1 Fj ✳ ◆♦t✐❝❡✱ t❤❛t ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ Fj ❛♥❞ Ek✱ ❢♦r

j ≥ n12

T❛♥❞ ✇r✐t✐♥❣ Aℓ ❢♦r Aℓ(T )✱ ✇❡ ❤❛✈❡

∑

ℓ∈[j2,(j+1)2] s✳t✳ ‖Aℓ‖...‖Aℓ+L‖>ℓξ

1

(ℓ)η≤

2L

j2η.

❍❡♥❝❡✱ s✐♥❝❡ η > 1/2 ∑

ℓ≥nTs✳t✳ ‖Aℓ‖...‖Aℓ+L‖>ℓξ

1

ℓη≤∑

j≥n12T

2

j2η< +∞.

❚❤❡r❡❢♦r❡ ❛♥❞ s✐♥❝❡ (qℓ)ℓ∈N ❣r♦✇s ❡①♣♦♥❡♥t✐❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✭❢♦r s♦♠❡ ❝♦♥st❛♥ts CT , cT > 0✮∑

ℓ∈N s✳t✳ ‖Aℓ‖...‖Aℓ+L‖>ℓξ

1

(log qℓ)η≤ CT +

∑

ℓ≥nTs✳t✳ ‖Aℓ‖...‖Aℓ+L‖>ℓξ

cTℓη

< +∞

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✷✺

❛♥❞ s♦ t❤❛t ❡q✉❛t✐♦♥ ✭✸✳✹✮ ✐♥ t❤❡ ❘❛t♥❡r ❉❈ ❤♦❧❞s ❢♦r ❛♥② T ∈ Y ′′✳

❚❤✉s✱ ✇❡ s❤♦✇❡❞ s♦ ❢❛r t❤❛t ❢♦r ❡✈❡r② T ∈ Y ′ ∩ Y ′′✱ ❛❧♦♥❣ t❤❡ s❡q✉❡♥❝❡ (nl)l ♦❢ r❡t✉r♥s t♦ Y ✱ ❜♦t❤

t❤❡ ▼✐①✐♥❣ ❉❈ ❢♦r T ∈MDC(τ, ℓ, ν

)❛♥❞ ✭✸✳✹✮ ✐♥ t❤❡ ❘❛t♥❡r ❉❈ ❤♦❧❞s✳ ❚❤✉s✱ s✐♥❝❡ µ(Y ′) = µ(Y ′′) =

µ(Y )✱ t❤❡ ❘❛t♥❡r ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✸✳✷✮ ❤♦❧❞s ❢♦r µ✲❛❧♠♦st ❡✈❡r② T ∈ Y ✳ ❙✐♥❝❡ ❜②

❘❡♠❛r❦ ✷✳✾ t❤✐s s❡q✉❡♥❝❡ ❞♦❡s ♥♦t ❡✈❡♥t✉❛❧❧② ❞❡♣❡♥❞ ♦♥ t❤❡ ❧✐❢t T = (τ, λ, π) ♦❢ t❤❡ ■❊❚ T = (λ, π)✱❜✉t ♦♥ T ♦♥❧② ❛♥❞ p∗µ = µ✱ ✐t ❢♦❧❧♦✇s t❤❛t t❤❡ s❡t Y ′ ⊂ Y ♦❢ ■❊❚s ❢♦r ✇❤✐❝❤ s✉❝❤ ❛ s❡q✉❡♥❝❡ ❡①✐sts ❤❛s♠❡❛s✉r❡ µ(Y ′) = Y ✳ ❈♦♥s✐❞❡r t❤❡ s❡t DCR ♦❢ T ∈ X s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts n✱ ❢♦r ✇❤✐❝❤ RnT ∈ Y ′✳ ❲❡❝❧❛✐♠ t❤❛t t❤✐s ✐s t❤❡ s❡t ♦❢ ❢✉❧❧ ♠❡❛s✉r❡ ♦❢ ■❊❚s ✇❤✐❝❤ s❛t✐s❢② t❤❡ ❞❡s✐r❡❞ ❘❛t♥❡r ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥✳❚♦ s❡❡ t❤❛t DCR ❤❛s ❢✉❧❧ ♠❡❛s✉r❡✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ✉s❡ ❡r❣♦❞✐❝✐t② ♦❢ Z ❛♥❞ t❤❡ ❢❛❝t t❤❛t µZ(Y

′) > 0✱r❡♠❛r❦✐♥❣ t❤❛t Z ♦r❜✐ts ❛r❡ s✉❜s❡ts ♦❢ R ♦r❜✐ts✳ ■❢ T ∈ DCR✱ t❤❡ s❡q✉❡♥❝❡ nl + nl + n✱ ✇❤❡r❡ nl ✐st❤❡ s❡q✉❡♥❝❡ ❛ss♦❝✐❛t❡❞ t♦ RnT ❝❧❡❛r❧② ❛❧s♦ s❛t✐s❢② t❤❡ ❘❛t♥❡r ❉❈ ❞❡✜♥✐t✐♦♥ ♣r♦♣❡rt②✳ ❋✐♥❛❧❧②✱ t❤❡❢♦r♠✉❧❛t✐♦♥ ✐♥ Pr♦♣♦s✐t✐♦♥ ✸✳✸ ❢♦❧❧♦✇s ❜② ❛❜s♦❧✉t❡ ❝♦♥t✐♥✉✐t② ♦❢ µZ ✇✳r✳t✳ ▲❡❜❡s❣✉❡✳ �

✹✳ ❇✐r❦❤♦❢❢ s✉♠s ♦❢ r♦♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st❛t❡ ♣r❡❝✐s❡ ❡st✐♠❛t❡s ♦♥ t❤❡ ❣r♦✇t❤ ♦❢ ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❢✉♥❝t✐♦♥s✇✐t❤ ❧♦❣❛r✐t❤♠✐❝ ❛s②♠♠♠❡tr✐❝ s✐♥❣✉❧❛r✐t✐❡s ✉♥❞❡r t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥s ✐♥tr♦❞✉❝❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳❚❤❡s❡ ❡st✐♠❛t❡s✱ ❛s ❡①♣❧❛✐♥❡❞ ✐♥ t❤❡ ♦✉t❧✐♥❡ ✐♥ ❙❡❝t✐♦♥ ✶✳✹✱ ❛r❡ ❛ ❝r✉❝✐❛❧ t♦♦❧ t♦ ♣r♦✈❡ ♠✐①✐♥❣ ❛♥❞♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♣❡❝✐❛❧ ✢♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✹✳✶ ✇❡ ✜rst r❡❝❛❧❧ ❛ ❝r✐t❡r✐✉♠ ✇❤✐❝❤❛❧❧♦✇s t♦ r❡❞✉❝❡ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❙❘✲♣r♦♣❡rt② t♦ ❛ st❛t❡♠❡♥t ❛❜♦✉t ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡ r♦♦❢ ❢✉♥❝t✐♦♥✳■♥ ❙❡❝t✐♦♥ ✹✳✷ ✇❡ st❛t❡ t❤❡ ❡st✐♠❛t❡s ♦♥ ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♣r♦✈❡❞ ✐♥ ❬✹✶✱ ✸✺❪ ✉♥❞❡r t❤❡▼✐①✐♥❣ ❉❈ ❛♥❞ ❞❡❞✉❝❡ ❡st✐♠❛t❡s ✐♥ ❢♦r♠ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❝♦♥✈❡♥✐❡♥t ❢♦r ✉s t♦ ♣r♦✈❡ t❤❡ ❙❲✲❘❛t♥❡r♣r♦♣❡rt② ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❋✐♥❛❧❧②✱ ✐♥ ❙❡❝t✐♦♥ ✹✳✸ ✇❡ s❤♦✇ t❤❛t t❤❡ ❘❛t♥❡r ❉❈ ❢♦r ❛ ❝❡rt❛✐♥ r❛♥❣❡ ♦❢♣❛r❛♠❡t❡rs ✐♠♣❧✐❡s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ s❡r✐❡s ✭s❡❡ t❤❡ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✐♥ ❉❡✜♥✐t✐♦♥ ✹✳✷✮ ✇❤✐❝❤✐s ✉s❡❢✉❧ ✇❤❡♥ ♣r♦✈✐♥❣ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡ ❡st✐♠❛t❡s✳

✹✳✶✳ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ❢♦r s♣❡❝✐❛❧ ✢♦✇s ✭♦✈❡r ■❊❚s✮ ✈✐❛ ❇✐r❦❤♦✛ s✉♠s✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ r❡❝❛❧❧❛ ❝r✐t❡r✐♦♥ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ ❙❘✲♣r♦♣❡rt② ✐♥ t❤❡ ❝❧❛ss ♦❢ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ❛♥ ❡r❣♦❞✐❝ ❛✉t♦♠♦r♣❤✐s♠✳■t ✇❛s st✉❞✐❡❞ ✐♥ ❬✶✵❪ ✐♥ t❤❡ ❝❛s❡ t❤❡ ❜❛s❡ ❛✉t♦♠♦r♣❤✐s♠ ✐s ❛♥ ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ❛♥❞✱ ✐♥ t❤❡ ❣❡♥❡r❛❧❝❛s❡ ✐♥ ❬✶✺❪✳

Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ▲❡t (X, d) ❜❡ ❛ σ✲❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✱ B t❤❡ σ✲❛❧❣❡❜r❛ ♦❢ ❇♦r❡❧ s✉❜s❡ts ♦❢ X✱ µ❛ ❇♦r❡❧ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ (X, d)✳ ▲❡t T ❜❡ ❛♥ ❡r❣♦❞✐❝ ❛✉t♦♠♦r♣❤✐s♠ ❛❝t✐♥❣ ♦♥ (X,B, µ) ❛♥❞ ❧❡t

f ∈ L1(X,B, µ) ❜❡ ❛ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ❜♦✉♥❞❡❞ ❛✇❛② ❢r♦♠ ③❡r♦✳ ▲❡t T = (T ft )t∈R ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣s♣❡❝✐❛❧ ✢♦✇✳ ▲❡t P = {−1, 1}✳ ❆ss✉♠❡ t❤❛t

❢♦r ❡✈❡r② ε > 0 ❛♥❞ N ∈ N t❤❡r❡ ❡①✐st κ = κ(ε) > 0✱ δ = δ(ε,N) > 0 ❛♥❞ ❛ s❡tX ′ = X ′(ε,N) ✇✐t❤ µ(X ′) > 1−ε✱ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x, y ∈ X ′ ✇✐t❤ 0 < d(x, y) < δt❤❡r❡ ❡①✐st M =M(x, y), L = L(x, y) ≥ N ✇✐t❤ L

M ≥ κ ❛♥❞ p = p(x, y) ∈ P

s✉❝❤ t❤❛t ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿

✭✐✮ d(Tnx, Tny) < ε ❛♥❞ |Sn(f)(x)− Sn(f)(y)− p| < ε ❢♦r ❡✈❡r② n ∈ [M,M + L]✱✭✐✐✮ d(T−nx, T−ny) < ε ❛♥❞ |S−n(f)(x)− S−n(f)(y)− p| < ε ❢♦r ❡✈❡r② n ∈ [M,M + L]✳

❚❤❡♥ T ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ ▲❡♠♠❛ ♣r♦✈✐❞❡s ❝♦♥❞✐t✐♦♥s t♦ ✈❡r✐❢②✐♥❣ ✭✐✮ ❛♥❞ ✭✐✐✮ ✐♥ Pr♦♣♦s✐t✐♦♥ ✹✳✶ ❛❜♦✈❡ ✐♥ ❝❛s❡ ♦❢s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s✳

▲❡♠♠❛ ✹✳✷✳ ▲❡t T : I → I ❜❡ ❛♥ ■❊❚✳ ❋✐① ε > 0 ❛♥❞ N ∈ N ❛♥❞ ❛ss✉♠❡ t❤❛t ❢♦r s♦♠❡ x, y ∈ I ✇✐t❤x < y✱ y − x < ε ❛♥❞ s♦♠❡ M,L ≥ N ✇❡ ❤❛✈❡✿

✭✹✳✶✮ {ℓα : α ∈ A} ∩ (

M+L⋃

i=0

T i[x, y]) = ∅,

✭✹✳✷✮[M,M + L]× [x, y] ∋ (n, θ) 7→ sign(Sn(f

′)(θ)) ✐s ❝♦♥st❛♥t ❢♦r ❡✈❡r② θ ∈ [x, y] ❛♥❞❡✈❡r② n ∈ [M,M + L]✱

✷✻ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

✭✹✳✸✮ (1− ε)(y − x)−1 < |Sn(f′)(θ)| < (1 + ε)(y − x)−1.

❚❤❡♥ x, y s❛t✐s❢② ✭✐✮ ✭✇✐t❤ M,L, ε✮✳ ❆♥❛❧♦❣♦✉s❧②✱ ✐❢

✭✹✳✹✮ {ℓα : α ∈ A} ∩ (M+L⋃

i=1

T−i[x, y]) = ∅,

✭✹✳✺✮[M,M +L]× [x, y] ∋ (n, θ) 7→ sign(S−n(f

′)(θ)) ✐s ❝♦♥st❛♥t ❢♦r ❡✈❡r② θ ∈ [x, y] ❛♥❞❡✈❡r② n ∈ [M,M + L]✱

✭✹✳✻✮ (1− ε)(y − x)−1 < |S−n(f′)(θ)| < (1 + ε)(y − x)−1,

t❤❡♥ x, y s❛t✐s❢② ✭✐✐✮✳

Pr♦♦❢✳ ❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ✭✹✳✶✮✱ ✭✹✳✷✮ ❛♥❞ ✭✹✳✸✮ ❤♦❧❞ ❛♥❞ s❤♦✇ ✭✐✮ ✭t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♦t❤❡r ♣❛rt ♦❢ t❤❡❛ss❡rt✐♦♥ ✐s ❛♥❛❧♦❣♦✉s✮✳ ◆♦t✐❝❡ ✜rst t❤❛t ❜② ✭✹✳✶✮✱ ❢♦r ❡✈❡r② n ∈ [M,M + L]✱

d(Tnx, Tny) = d(x, y) = y − x < ε,

s♦ t❤❡ ✜rst ♣❛rt ♦❢ ✭✐✮ ❤♦❧❞s✳ ▼♦r❡♦✈❡r✱ ❜② ✭✹✳✶✮ ❛♥❞ t❤❡ ❢❛❝t t❤❛t f ∈ C2(T \ {ℓα : α ∈ A})✱ ❢♦r ❡✈❡r②n ∈ [M,M + L]✱ ✇❡ ❤❛✈❡

✭✹✳✼✮ Sn(f)(x)− Sn(f)(y) = (x− y)Sn(f′)(θn) ❢♦r s♦♠❡ θn ∈ [x, y].

❇② ✭✹✳✷✮✱ ✇❡ ❝❛♥ ❛ss✉♠❡ ❲▲❖● t❤❛t ❢♦r ❡✈❡r② n ∈ [M,M + L] ❛♥❞ ❡✈❡r② θ ∈ [x, y]✱ Sn(f′)(θn) > 0

✭t❤❡ ♦♣♣♦s✐t❡ ❝❛s❡ ✐s ❛♥❛❧♦❣♦✉s✮✳ ❚❤❡♥✱ ❜② ✭✹✳✼✮✱ ❢♦r ❡✈❡r② n ∈ [M,M + L]✱ ✉s✐♥❣ ✭✹✳✸✮✱ ✇❡ ♦❜t❛✐♥

|Sn(f)(x)− Sn(f)(y) + 1| = |(y − x)Sn(f′)(θn)− 1| ≤ ε,

s♦ ✭✐✮ ❤♦❧❞s ✇✐t❤ p = −1 ∈ P ✳ �

✹✳✷✳ ●r♦✇t❤ ♦❢ ❇✐r❦❤♦✛ s✉♠s ♦❢ ❞❡r✐✈❛t✐✈❡s✳ ❚❤r♦✉❣❤♦✉t t❤✐s s❡❝t✐♦♥✱ ❧❡t T ❜❡ ❛♥ ■❊❚ ❛♥❞ ❧❡tf ∈ AsymLogSing(T )✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❛ss✉♠❡ t❤❛t ■❊❚ T s❛t✐s✜❡s ▼✐①✐♥❣ ❉❈✳ ❇❡❢♦r❡ st❛t✐♥❣ q✉❛♥t❛t✐✈❡r❡s✉❧ts ♦♥ t❤❡ ❣r♦✇t❤ ♦❢ t❤❡ ❇✐r❦❤♦✛ s✉♠s Sr(f

′′)(x) ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ f ′ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦✈❡r T ✱ ✇❡ ✇✐❧❧✐♥tr♦❞✉❝❡ s♦♠❡ ♥♦t❛t✐♦♥ ❛♥❞ ❞❡✜♥✐t✐♦♥s✳

❉❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ {σℓ}ℓ∈N✱ ✉s❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ❜❡❧♦✇ ❛s ❛ t❤r❡s❤♦❧❞t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r r ✐s ❝❧♦s❡r t♦ ql ♦r t♦ ql+1✳ ▲❡t τ ′ ❜❡ s✉❝❤ t❤❛t τ/2 < τ ′ < 1✱ ✇❤❡r❡ τ ✐s t❤❡❉✐♦♣❤❛♥t✐♥❡ ❡①♣♦♥❡♥t ✐♥ ✭✸✳✶✮ ❣✐✈❡♥ ❜② Pr♦♣♦s✐t✐♦♥ ✸✳✸ ❛♥❞ τ ′ ✐s ✇❡❧❧ ❞❡✜♥❡❞ s✐♥❝❡ τ < 2✳ ▲❡t

✭✹✳✽✮ σℓ = σℓ(T ) +

(log ‖Aℓ‖

log qℓ

)τ ′,

τ

2< τ ′ < 1,

❈❧❡❛r❧② σℓ ❞❡♣❡♥❞s ♦♥ t❤❡ ■❊❚ T ✇❡ st❛rt ✇✐t❤✱ s✐♥❝❡ t❤❡ s❡q✉❡♥❝❡ {nℓ}ℓ∈N ❞♦❡s✳❚❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ✐s t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤✐❝❤ ♦♥❡ ♥❡❡❞s t♦ t❤r♦✇ ❛✇❛② ✐♥ ♦r❞❡r t♦ ❣❡t ❡st✐♠❛t❡s ♦♥

❇✐r❦❤♦✛ s✉♠s ❢♦r t✐♠❡s ❜❡t✇❡❡♥ qℓ ❛♥❞ qℓ+1 ✭s❡❡ Pr♦♣♦s✐t✐♦♥ ✹✳✹✮✳ ❚❤❡s❡ ❛r❡ ♣♦✐♥ts t❤❛t ❣❡t t♦♦ ❝❧♦s❡t♦ t❤❡ s✐♥❣✉❧❛r✐t✐❡s✱ s♦ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❇✐r❦❤♦✛ s✉♠s ❝❛♥ ❜❡ ❛r❜✐tr❛r✐❧② ❧❛r❣❡✳

❉❡✜♥✐t✐♦♥ ✹✳✶✳ ▲❡t Σ+ℓ = Σ+

ℓ (T ) ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t✱ ✇❤❡r❡ [·] ❞❡♥♦t❡s t❤❡ ❢r❛❝t✐♦♥❛❧ ♣❛rt✿

✭✹✳✾✮ Σ+ℓ (T ) +

⋃

α∈A

[σℓqℓ+1]⋃

i=0

T−i[−σℓI(nℓ) + ℓα, lα + σℓI

(nℓ)].

❘❡♠❛r❦ ✹✳✸✳ ◆♦t✐❝❡ t❤❛t s✐♥❝❡ nℓ ✐s ❛ ν✲❜❛❧❛♥❝❡❞ t✐♠❡ ✭s❡❡ ❘❡♠❛r❦ ✷✳✶✶✮

λ(Σ+ℓ ) ≤ 2|A|νσ2ℓ

qℓ+1

qℓ≤ 2|A|ν2σ2ℓ ‖Aℓ‖.

❋♦r α ∈ A✱ ❧❡t ✉s ❞❡♥♦t❡ ❜② uα(x) =1

x−rα❛♥❞ ❜② vα(x) =

1lα−x

✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s U(r, x)

❛♥❞ V (r, x) ❣✐✈❡ t❤❡ ❧❛r❣❡st ❝♦♥tr✐❜✉t✐♦♥ ✐♥ ❛ ❇✐r❦❤♦✛ s✉♠ ♦❢ ❧❡♥❣t❤ r st❛rt✐♥❣ ❢r♦♠ x ❣✐✈❡♥ r❡s♣❡❝t✐✈❡❧②t♦ ✈✐s✐ts ♦❢ t❤❡ ♦r❜✐t ✇❤✐❝❤ ❛r❡ ❝❧♦s❡ t♦ ❛ s✐♥❣✉❧❛r✐t② ❢♦r♠ t❤❡ r✐❣❤t ♦r ❢r♦♠ t❤❡ ❧❡❢t r❡s♣❡❝t✐✈❡❧②✿

U(r, x) = maxα∈A

max0≤i<r

uα(Tix), V (r, x) = max

α∈Amax0≤i<r

vα(Tix).

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✷✼

❋♦r ♣♦✐♥ts ♦✉ts✐❞❡ t❤❡ s❡t σℓ ♦♥❡ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r❡❝✐s❡ ❡st✐♠❛t❡s✱ ✇❤✐❝❤ ✇❡r❡ ♣r♦✈❡❞ ❜② t❤❡ t❤✐r❞❛✉t❤♦r ✐♥ ❬✹✶❪ ❢♦r f ✇✐t❤ ♦♥❡ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t② ❛♥❞ t❤❡♥ ❡①t❡♥❞❡❞ ❜② ❘❛✈♦tt✐ ✐♥ ❬✸✺❪t♦ ❣❡♥❡r❛❧ f ∈ AsymLog(T )✳

Pr♦♣♦s✐t✐♦♥ ✹✳✹ ✭●r♦✇t❤s ♦❢ ❞❡r✐✈❛t✐✈❡s✱ ❬✹✶✱ ✸✺❪✮✳ ▲❡t T ❜❡ ❛♥ ■❊❚ ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ▼✐①✐♥❣ ❉❈ ❛❧♦♥❣❛ s✉❜s❡q✉❡♥❝❡ (nℓ) ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s✳ ▲❡t f ∈ AsymLogSing(T )✳ ❆ss✉♠❡ ❲▲❖● t❤❛t C− > C+✳❚❤❡r❡ ❡①✐sts M > 0 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ε > 0 t❤❡r❡ ❡①✐sts ℓ1 ∈ N s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ℓ ≥ ℓ1

qℓ ≤ r < qℓ+1 ❛♥❞ x /∈ Σ+ℓ ,

✇❡ ❤❛✈❡

(C− − C+ − ε2)r log r ≤ Sr(f′)(x) ≤ (C− − C+ + ε2)r log r +M(U(r, x) + V (r, x)).

❲❡ ♣r❡s❡♥t ♥♦✇ s♦♠❡ ❡st✐♠❛t❡s ♦♥ ❇✐r❦❤♦✛ s✉♠s ✭▲❡♠♠❛ ✹✳✻✮ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ❜② Pr♦♣♦s✐t✐♦♥✹✳✹ ❛♥❞ ❛r❡ ❣✐✈❡♥ ✐♥ ❛ ❢♦r♠ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❝♦♥✈❡♥✐❡♥t ❢♦r ✉s t♦ ♣r♦✈❡ t❤❡ ❙❲✲❘❛t♥❡r ♣r♦♣❡rt②✳ ■♥ ♦r❞❡rt♦ ♣r♦✈❡ t❤❡ q✉❛♥t✐t❛t✐✈❡ ❡st✐♠❛t❡s ♦♥ ❇✐r❦❤♦✛ s✉♠s ✐♥ ▲❡♠♠❛ ✹✳✻✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡❝❤♥✐❝❛❧▲❡♠♠❛✳

▲❡♠♠❛ ✹✳✺✳ ■❢ τ, τ ′, ξ, η ❛r❡ s✉❝❤ t❤❛t

τ ∈ (1, 16/15) , τ ′ ∈ (15/16, 1) , η ∈(3/4, 2τ ′ − τ

), ξ ∈

(max(11/12, τ ′η), τ ′

),

t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✿

✭✹✳✶✵✮ limℓ→+∞

σℓ(log qℓ)ξ = 0;

✭✹✳✶✶✮ limℓ→+∞

σ2−ηℓ ℓτ = 0;

✭✹✳✶✷✮ limℓ→+∞

log ‖Aℓ‖

(log qℓ)ξσηℓ

= 0;

Pr♦♦❢✳ ◆♦t✐❝❡ ✜rst t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c > 0 s✉❝❤ t❤❛t

✭✹✳✶✸✮ qℓ ≥ cℓ.

❋♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ℓ ∈ N✱ ❜② t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✸✳✶✱ ‖Aℓ‖ ≤ ℓτ ✳ ❙✐♥❝❡ ξ < τ ′ ✇❡ ❣❡t ✭s❡❡ ✭✹✳✽✮✮

σℓ(log qℓ)ξ ≤

τ log ℓ

(log qℓ)τ′−ξ

.

❚❤✐s ❛♥❞ ✭✹✳✶✸✮ ❣✐✈❡ ✭✹✳✶✵✮✳ ❚♦ ♣r♦✈❡ ✭✹✳✶✶✮ ♥♦t✐❝❡ t❤❛t ❜② ✭✹✳✽✮✱ t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✭✸✳✶✮ ❛♥❞✭✹✳✶✸✮ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ℓ ✇❡ ❤❛✈❡ ✭❢♦r s♦♠❡ ❝♦♥st❛♥t C > 0✮

✭✹✳✶✹✮ σ2−ηℓ ℓτ ≤ C(τ log ℓ)(2−η)τ

′

ℓ(2−η)τ ′−τ.

❙✐♥❝❡ ητ ′ < η < 2τ ′ − τ ❜② t❤❡ ❛ss✉♠♣t✐♦♥s ♦♥ η, τ, τ ′✱ ✇❡ ❤❛✈❡ t❤❛t (2 − η)τ ′ − τ > 0✳ ❚❤✉s✱ ✭✹✳✶✶✮❢♦❧❧♦✇s ❢r♦♠ ✭✹✳✶✹✮✳ ◆❡①t✱ ♥♦t✐❝❡ t❤❛t

log ‖Aℓ‖

(log qℓ)ξσηℓ

≤τ log ℓ

(log qℓ)ξ−ητ′,

t❤✐s ✜♥✐s❤❡s t❤❡ ♣r♦♦❢ ♦❢ ✭✹✳✶✷✮ s✐♥❝❡✱ ❜② ❛ss✉♠♣t✐♦♥s✱ ξ − ητ ′ > 0 ❛♥❞ ✭✹✳✶✸✮ ❤♦❧❞s✳ �

❆ss✉♠♣t✐♦♥✳ ❋r♦♠ ♥♦✇ ♦♥ ✇❡ ♠❛❦❡ ❛ st❛♥❞✐♥❣ ❛ss✉♠♣t✐♦♥✼ ♦♥ τ, τ ′, ξ, η✱ ♥❛♠❡❧②

✭✹✳✶✺✮ τ ∈

(1,

16

15

), τ ′ ∈

(15

16, 1

), η ∈

(3/4, τ ′(2τ ′ − τ)

), ξ ∈

(max(11/12, τ ′η), τ ′

).

❖♥❡ ❝❛♥ ✈❡r✐❢② t❤❛t ❛❧❧ ✐♥t❡r✈❛❧s ❛r❡ ✐♥❞❡❡❞ ♥♦♥✲❡♠♣t②✱ s♦ t❤❛t s✉❝❤ ❛ ❝❤♦✐❝❡ ❡①✐sts✳ ◆❡①t ❧❡♠♠❛ ❛❧❧♦✇s✉s t♦ ❝♦♥tr♦❧ ❢♦r✇❛r❞ ✭❜❛❝❦✇❛r❞✮ ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢♦r ♣♦✐♥ts ✇❤♦s❡ ❢♦r✇❛r❞s ✭❜❛❝❦✇❛r❞s✮

✼❲❡ r❡♠❛r❦ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ η ✇❡ ❛s❦ ❢♦r ❤❡r❡ ✐s ♦♥ ♣✉r♣♦s❡ ♠♦r❡ r❡str✐❝t✐✈❡ ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥ r❡q✉✐r❡❞ ✐♥ ▲❡♠♠❛✹✳✺✱ s✐♥❝❡ t❤✐s ✇✐❧❧ ❜❡ ✉s❡❢✉❧ ❧❛t❡r✳

✷✽ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

♦r❜✐ts ❞♦ ♥♦t ❝♦♠❡ t♦ ❝❧♦s❡ t♦ s✐♥❣✉❧❛r✐t✐❡s✳ ■t s❛②s t❤❛t ✐❢ ♦r❜✐t ♦❢ ❛ ♣♦✐♥t st❛②s ❛✇❛② ❢r♦♠ s✐♥❣✉❧❛r✐t②✱t❤❡♥ ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ✭✉♣ t♦ ❛ s♠❛❧❧ ❡rr♦r✮ ❡q✉❛❧ t♦ t❤❡ ♠❛✐♥ ❝♦♥tr✐❜✉t✐♦♥ ✭❝♦♠♠✐♥❣❢r♦♠ s✉♠s ❛❧♦♥❣ t❤❡ ♦r❜✐t✮✳ ❚❤❡ ♠❛✐♥ t♦♦❧ ✐s Pr♦♣♦s✐t✐♦♥ ✹✳✹✳

▲❡♠♠❛ ✹✳✻ ✭●r♦✇t❤ ♦❢ ❞❡r✐✈❛t✐✈❡s ❢♦r ❙❘✲♣r♦♣❡rt②✮✳ ❋♦r ❡✈❡r② ε > 0 t❤❡r❡ ❡①✐sts ℓ1 ∈ N s✉❝❤ t❤❛t ❢♦r❡✈❡r② ℓ ≥ ℓ1 ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✭❆✮ ✐❢

✭✹✳✶✻✮ U(qℓ+1, x), V (qℓ+1, x) ≤ 2qℓ(log qℓ)ξ,

t❤❡♥ ❢♦r qℓ ≤ r < qℓ+1 ✇❡ ❤❛✈❡

✭✹✳✶✼✮ (C− − C+ − ε2)r log r ≤ Sr(f′)(x) ≤ (C− − C+ + ε2)r log r;

✭❇✮ ✐❢

✭✹✳✶✽✮ U(qℓ+1, T−qℓ+1x), V (qℓ+1, T

−qℓ+1x) ≤ 2qℓ(log qℓ)ξ,

t❤❡♥ ❢♦r h(nl) ≤ r < h(nl+1) ✇❡ ❤❛✈❡

✭✹✳✶✾✮ (C− − C+ − ε2)r log r ≤ −S−r(f′)(x) ≤ (C− − C+ + ε2)r log r.

Pr♦♦❢✳ ❋✐① ε > 0✳ ▲❡t ✉s ✜rst s❤♦✇ ✭❆✮✳ ❋r♦♠ ▲❡♠♠❛ ✹✳✺✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ℓ ✭❞❡♣❡♥❞✐♥❣ ♦♥ ε✮ ✇❡❤❛✈❡

✭✹✳✷✵✮ (log qℓ)ξ <

ε

σℓ

▲❡t M ❜❡ t❤❡ ❝♦♥st❛♥t ❣✐✈❡♥ ❜② Pr♦♣♦s✐t✐♦♥ ✹✳✹✳ ◆♦t✐❝❡ t❤❛t ❢♦r ❡✈❡r② i ∈ [0, σℓqℓ+1] ⊂ [0, qℓ+1] ❛♥❞❡✈❡r② α ∈ A✱ ✉s✐♥❣ ✭✹✳✶✻✮ ❛♥❞ ✭✹✳✷✵✮ ✭s✐♥❝❡ nℓ ✐s ❛ ❜❛❧❛♥❝❡❞ t✐♠❡ ❛♥❞ ε ✐s s♠❛❧❧✮

d(T ix, lα) ≥1

2qℓ(log qℓ)ξ≥ σℓI

(nℓ).

❚❤❡r❡❢♦r❡ x /∈ Σ+ℓ ✭s❡❡ ✭✹✳✾✮✮✳ ▼♦r❡♦✈❡r✱ ✐❢ ℓ ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ ✭s✐♥❝❡ ξ < 1✮✱ ✉s✐♥❣ ✭✹✳✶✻✮✱ ✇❡ ❤❛✈❡ ❢♦r

❡✈❡r② r ∈ [qℓ, qℓ+1)✱

✭✹✳✷✶✮ε

2Mr log r ≥

ε

2Mqℓ log qℓ ≥ 2qℓ(log qℓ)

ξ ≥ U(qℓ+1, x) ≥ U(r, x),

✭t❤❡ s❛♠❡ ❡st✐♠❛t❡s ❤♦❧❞ ❢♦r V (r, x)✮✳❙✐♥❝❡ x /∈ Σ+

ℓ ✱ ✇❡ ❝❛♥ ✉s❡ ❡st✐♠❛t❡s ✐♥ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ❢♦r ε2 ❛♥❞ ✉s✐♥❣ ✭✹✳✷✶✮ ✇❡ ❦♥♦✇ t❤❛t ❢♦r ℓ ≥ ℓ1

✭ℓ1 ❞❡♣❡♥❞✐♥❣ ♦♥ ε✮✱

(C− − C+ − ε)r log r ≤ Sr(f′)(x) ≤ (C− − C+ + ε)r log r.

❚❤✐s ❣✐✈❡s ✭✹✳✶✼✮✳◆♦✇ ✇❡ s❤♦✇ ✭✹✳✶✾✮✳ ❋✐① r ∈ [qℓ, qℓ+1)✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t T−rx /∈ Σ+

ℓ ✳ ❋♦r t❤✐s ❛✐♠ ♥♦t❡ ✜rst t❤❛t✐❢ ℓ ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡✱ t❤❡♥

✭✹✳✷✷✮ σℓqℓ+1 < qℓ.

■♥❞❡❡❞✱ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ qℓ+1 ✐t ❢♦❧❧♦✇s t❤❛t ❢♦r ❡✈❡r② z ∈ T✱

✭✹✳✷✸✮ {z, . . . , T qℓ+1z} ∩ Inℓ+1 6= ∅.

■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r z = T−qℓ+1x✱ ❜② t❤❡ ❢❛❝t t❤❛t nℓ+1 ✐s ❛ ❜❛❧❛♥❝❡❞ t✐♠❡✱ ✭✹✳✷✸✮ ❛♥❞ ✭✹✳✶✽✮✱ ✐t ❢♦❧❧♦✇st❤❛t

νqℓ+1 ≤1

I(nℓ+1)≤ U(qℓ+1, T

−qℓ+1x) ≤ 2qℓ(log qℓ)ξ.

❚❤✐s ❛♥❞ ✭✹✳✷✵✮ ❣✐✈❡s ✭✹✳✷✷✮ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ℓ✳ ❇② ✭✹✳✷✷✮ ✐t ❢♦❧❧♦✇s t❤❛t

✭✹✳✷✹✮ Σ+ℓ ⊂

⋃

α∈A

qℓ−1⋃

i=0

T−i[−σℓI(nℓ) + lα, lα + σℓI

(nℓ)].

◆♦t✐❝❡ ❤♦✇❡✈❡r✱ t❤❛t ❜② ✭✹✳✶✽✮✱ ❢♦r ❡✈❡r② i ∈ [0, qℓ) ❛♥❞ ❡✈❡r② α ∈ A ✭qℓ+1 > r ≥ qℓ✮✱ ✇❡ ❤❛✈❡

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✷✾

d(T i(T−rx), lα) = d(T i−rx, lα) ≥ sup−qℓ+1<s<0

d(T sx, lα) ≥

1

max(U(qℓ+1, T−qℓ+1x), V (qℓ+1, T−qℓ+1x))≥

1

2qℓ(log qℓ)ξ≥ σℓI

(nℓ),

t❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ❜② ✭✹✳✷✵✮ ✭❛♥❞ ❜❛❧❛♥❝❡✮✳ ❚❤❡r❡❢♦r❡✱

T−rx /∈⋃

α∈A

qℓ−1⋃

i=0

T−i[−σℓI(nℓ) + lα, lα + σℓI

(nℓ)],

❛♥❞ ❜② ✭✹✳✷✹✮✱ T−rx /∈ Σ+ℓ ✳ ◆♦t✐❝❡ t❤❛t

✭✹✳✷✺✮ U(r, T−rx) ≤ U(qℓ+1, T−qℓ+1x) ❛♥❞ V (r, T−rx) ≤ V (qℓ+1, T

−qℓ+1x).

▼♦r❡♦✈❡r✱ ❜② t❤❡ ❝♦❝②❝❧❡ ✐❞❡♥t✐t② −S−r(f′)(x) = Sr(f

′)(T−rx)✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ✉s❡ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ❢♦rr ❛♥❞ T−rx ❛♥❞ ✉s❡ ✭✹✳✷✺✮ ❛♥❞ ✭✹✳✶✽✮✱ t♦ ❣❡t

(C− − C+ − ε)r log r ≤ Sr(f′)(T−rx) ≤ (C− − C+ + ε)r log r.

❚❤✐s ✜♥✐s❤❡s t❤❡ ♣r♦♦❢ ♦❢ ✭✹✳✶✾✮✳ �

✹✳✸✳ ❘❛t♥❡r ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❞❡❞✉❝❡ ❢r♦♠ t❤❡ ❘❛t♥❡r ❉❈ ❛ s✉♠♠❛✲❜✐❧✐t② ❝♦♥❞✐t✐♦♥✱ ✇❤✐❝❤ ✐s ✈❡r② ❝♦♥✈❡♥✐❡♥t ✇❤❡♥ st✉❞②✐♥❣ ❇✐r❦❤♦✛ s✉♠s s✐♥❝❡ ✐t ❛❧❧♦✇s t♦ ❡st✐♠❛t❡ t❤❡♠❡❛s✉r❡ ♦❢ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤✐❝❤ ✇❡ ♥❡❡❞ t♦ t❤r♦✇ ❛✇❛② t♦ ❣❡t t❤❡ ❝♦♥tr♦❧ ♦❢ t❤❡ ❣r♦✇t❤ ♦❢ ❇✐r❦❤♦✛s✉♠s t♦ ♣r♦✈❡ t❤❡ ❙❘✲❝♦♥❞✐t✐♦♥ ✭s❡❡ t❤❡ ❤❡✉r✐st✐❝ ❞✐s❝✉ss✐♦♥ ✐♥ t❤❡ ♦✉t❧✐♥❡ ✐♥ ❙❡❝t✐♦♥ ✶✳✹✮✳

❉❡✜♥✐t✐♦♥ ✹✳✷✳ ❲❡ s❛② t❤❛t ❛♥ ■❊❚ T t❤❛t s❛t✐s✜❡s t❤❡ ♠✐①✐♥❣ ❉❈ ✇✐t❤ ♣♦✇❡r τ s❛t✐s✜❡s t❤❡ ❘❛t♥❡r❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✭♦r ❢♦r s❤♦rt✱ t❤❡ s✉♠♠❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥✮ ✇✐t❤ ❡①♣♦♥❡♥ts (τ ′, ξ′, η′) ✐❢

✭✹✳✷✻✮∑

ℓ/∈KT

ση′

ℓ < +∞, ✇❤❡r❡ KT +

{ℓ ∈ N : qℓ+L ≤

qℓ

σξ′

ℓ

}.

❲❡ r❡♠❛r❦ t❤❛t t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ τ ′ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s t❤r♦✉❣❤ σℓ ✇❤✐❝❤ ❛♣♣❡❛rs ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥♦❢ KT ✱ s✐♥❝❡ ❛s ♦♥❡ ❝❛♥ s❡❡ ✐♥ ✭✹✳✽✮✱ σℓ ❞❡♣❡♥❞s ♦♥ τ

′✳

▲❡♠♠❛ ✹✳✼✳ ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ✭✹✳✶✺✮ ♦♥ t❤❡ ♣❛r❛♠❡t❡rs τ, τ ′, ξ, η✱ ✐❢ T ∈ RDC (τ, ξ, η)✱ ❢♦r ❛♥②

ξ′ > ξτ ′ ❛♥❞ η′ > η

τ ′ ✱ T s❛t✐s✜❡s t❤❡ ❘❛t♥❡r ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✇✐t❤ ❡①♣♦♥❡♥ts (τ ′, ξ′, η′)✳

Pr♦♦❢✳ ◆♦t✐❝❡ ✜rst t❤❛t✱ s✐♥❝❡ log qℓ ≥ cℓ ✭❢♦r s♦♠❡ ❝♦♥st❛♥t c > 0✮✱ ❢♦r ❛♥② ǫ > 0✱ log ‖Aℓ‖/(log qℓ)ǫ

t❡♥❞s t♦ ③❡r♦ ❛s ℓ ❣r♦✇s ❜② t❤❡ ✐♥t❡❣r❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✭✸✳✶✮✳ ❚❤✉s✱ r❡❝❛❧❧✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✹✳✽✮ ♦❢ σℓ✱✇❡ ❤❛✈❡ t❤❛t ❢♦r ❛♥② τ ′′ < τ ′✱ ❢♦r ❛♥② ℓ s✉✣❝✐❡♥t❧② ❧❛r❣❡✱

✭✹✳✷✼✮ σℓ =(log ‖Aℓ‖)

τ ′

(log qℓ)τ′−τ ′′

1

(log qℓ)τ′′≤

1

(log qℓ)τ′′.

◆♦t✐❝❡ t❤❛t ξ′ > ξ′τ ′+ξ2 > ξ✳ ❚❤❡r❡❢♦r❡ ❛♥❞ ❜② ✭✹✳✷✼✮ ✇❡ ❤❛✈❡ ✭❢♦r ℓ s✉✣❝✐❡♥t❧② ❧❛r❣❡✮✱ ✇❡ ❤❛✈❡

1

σξ′

ℓ

≥ (log qℓ)ξ′τ ′+ξ

2 > ℓξ.

❚❤❡r❡❢♦r❡✱ ✐❢ ℓ /∈ KT ✱ t❤❡♥✱ ✇r✐t✐♥❣ Aℓ ❢♦r Aℓ(T ) ❛♥❞ t❛❦✐♥❣ L ❛s ✐♥ ✭✸✳✹✮✱ ✇❡ ❤❛✈❡ t❤❛t

✭✹✳✷✽✮ ‖Aℓ‖‖Aℓ+1‖ · · · ‖Aℓ+L‖ ≥qℓ+Lqℓ

>1

σξ′

ℓ

> ℓξ.

▼♦r❡♦✈❡r✱ s✐♥❝❡ η′τ ′ > η✱ t❤❛♥❦s t♦ ✭✹✳✷✼✮ ✇❡ ❛❧s♦ ❤❛✈❡ t❤❛t ση′

ℓ < 1/(log qℓ)η✳ ❍❡♥❝❡

✭✹✳✷✾✮∑

ℓ/∈KT

ση′

ℓ <∑

ℓ∈N s✳t✳ ‖Aℓ‖‖Aℓ+1‖···‖Aℓ+L‖>ℓξ

ση′

ℓ <∑

ℓ∈N s✳t✳ ‖Aℓ‖‖Aℓ+1‖···‖Aℓ+L‖>ℓξ

1

(log qℓ)η< +∞.

�

✸✵ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

❚❤❡ ❘❛t♥❡r ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥✱ ❛s ❛ ❝♦r♦❧❧❛r② ♦❢ ▲❡♠♠❛ ✹✳✺✱ ✐♠♣❧✐❡s ✐♥ ♣❛rt✐❝✉❧❛r t❤❛t t❤❡ s❡r✐❡s♦❢ ♠❡❛s✉r❡s ♦❢ t❤❡ s❡ts Σ+

ℓ (T ) ♦❢ ♣♦✐♥ts ♥♦t ❝♦♥tr♦❧❧❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✹✳✾✮ ✐s s✉♠♠❛❜❧❡

✐❢ ♦♥❡ r❡str✐❝ts t❤❡ s✉♠ ♦♥❧② t♦ ℓ ∈ N s✉❝❤ t❤❛t ‖Aℓ‖‖Aℓ+1‖ · · · ‖Aℓ+L‖ > ℓξ✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ✉s❡❢✉❧ t♦♣r♦✈❡ t❤❡ ❙❘✲❝♦♥❞✐t✐♦♥ ✭s❡❡ t❤❡ ♦✉t❧✐♥❡ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✮✳

❈♦r♦❧❧❛r② ✹✳✽✳ ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ▲❡♠♠❛ ✹✳✼✱ ✇❡ ❤❛✈❡ t❤❛t

✭✹✳✸✵✮∑

ℓ/∈KT

λ(Σ+ℓ (T )

)≤∑

ℓ/∈KT

λ(Σ+ℓ (T )

)<∞,

✇❤❡r❡ KT ✐s ❞❡✜♥❡❞ ❛s ✐♥ ✭✹✳✷✻✮ ❛♥❞ KT + {ℓ ∈ N : ‖Aℓ‖‖Aℓ+1‖ · · · ‖Aℓ+L‖ ≤ ℓξ}✳

Pr♦♦❢✳ ❚❤❡ ✐♥❡q✉❛❧✐t② ❜❡t✇❡❡♥ t❤❡ t✇♦ s❡r✐❡s ✐♥ ✭✹✳✸✵✮ ❢♦❧❧♦✇s ❢r♦♠ ✭✹✳✷✽✮✱ ✇❤✐❝❤ s❤♦✇s t❤❛t ✐❢ ℓ /∈ KT

t❤❡♥ ℓ /∈ KT ✳ ▲❡t ✉s ❝❤♦♦s❡ η′ s✉❝❤ t❤❛t✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ η′ > η/τ ′✱ ✐t ❛❧s♦ s❛t✐s✜❡s η′ ∈ (3/4, 2τ ′ − τ)✱

✇❤✐❝❤ ✐s ♣♦ss✐❜❧❡ s✐♥❝❡ η/τ ′ < 2τ ′ − τ ❜② t❤❡ ❝❤♦✐❝❡ ♦❢ η ✭r❡❝❛❧❧ ✭✹✳✶✺✮✮✳ ❚❤❡♥✱ ❜② ✭✹✳✶✶✮ ✐♥ ▲❡♠♠❛ ✹✳✺

❛♣♣❧✐❡❞ t♦ η′✱ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ℓ ✇❡ ❤❛✈❡ σ2ℓ ℓτ ≤ ση

′

ℓ /2✳ ❚❤❡r❡❢♦r❡✱ s✐♥❝❡ ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ℓ ✇❡❛❧s♦ ❤❛✈❡ t❤❛t ‖Aℓ‖ ≤ ℓτ ❜② t❤❡ ▼✐①✐♥❣ ❉❈ ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✸✳✶✮✱ ❜② ❘❡♠❛r❦ ✹✳✸ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥tC > 0 s✉❝❤ t❤❛t ∑

ℓ/∈KT

λ (Σℓ(T )) ≤∑

ℓ/∈KT

2|A|ν2σ2ℓ ‖Aℓ‖ ≤ C∑

ℓ/∈KT

ση′

ℓ ,

✇❤✐❝❤ ✐s ✜♥✐t❡ ❜② ✭✹✳✷✾✮✳ �

❖♥❡ ❝❛♥ s❤♦✇ ❛s ❛ ❝♦r♦❧❧❛r② ♦❢ ▲❡♠♠❛ ✹✳✼ t❤❛t ❢♦r ❛ s✉✐t❛❜❧② ❝❤♦s❡♥ r❛♥❣❡ ♦❢ ❡①♣♦♥❡♥ts τ ′✱ξ′ ❛♥❞ η′

t❤❡ s❡t ♦❢ ■❊❚✬s s❛t✐s❢②✐♥❣ t❤❡ ❙✉♠♠❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✇✐t❤ ❡①♣♦♥❡♥ts (τ ′, ξ′, η′) ❤❛s ❢✉❧❧ ♠❡❛s✉r❡✳

❈♦r♦❧❧❛r② ✹✳✾✳ ▲❡t τ ∈ (1, 16/15)✱ τ ′ ∈ (15/16, 1)✱ ξ′ > 99/100✱ η′ > 3/4✳ ❋♦r ❡❛❝❤ ✐rr❡❞✉❝✐❜❧❡❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t✉♠ π ❛♥❞ ❢♦r ▲❡❜❡s❣✉❡ ❛✳❡✳ λ ∈ ∆d✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ■❊❚ T = (λ, π) s❛t✐s✜❡s t❤❡❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✇✐t❤ ❡①♣♦♥❡♥ts (τ ′, ξ′, η′)✳

Pr♦♦❢✳ ❚❛❦❡ ❛♥② ξ, η s✉❝❤ t❤❛t 11/12 < ξ < ξ′τ ′ ❛♥❞ 1/2 < η < η′τ ′✱ ✇❤✐❝❤ ✐s ♣♦ss✐❜❧❡ ❜② t❤❡ ❛ss✉♠♣t✐♦♥s♦♥ t❤❡ ♣❛r❛♠❡t❡rs✳ ❈♦♥s✐❞❡r ❛♥② ■❊❚ T ✐♥ RCD(τ, ξ, η)✳ ❚❤❡ s❡t ♦❢ s✉❝❤ T ❤❛s ❢✉❧❧ ♠❡❛s✉r❡✱ s❡❡Pr♦♣♦s✐t✐♦♥ ✸✳✻✳ ❚❤❡♥✱ ❜② ▲❡♠♠❛ ✹✳✼✱ T s❛t✐s✜❡s t❤❡ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✇✐t❤ ❡①♣♦♥❡♥ts (τ ′, ξ′, η′)✳❚❤✐s ✜♥✐s❤❡s t❤❡ ♣r♦♦❢✳ �

❘❡♠❛r❦ ✹✳✶✵✳ ❲❤✐❧❡ t❤❡ ❘❛t♥❡r ❉❈ ❛s ❢♦r♠✉❧❛t❡❞ ✐s ✉s❡❢✉❧ ✇❤❡♥ tr②✐♥❣ t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ ■❊❚ss❛t✐s❢②✐♥❣ t❤❡ ❘❛t♥❡r ❉❈ ❢♦r s✉✐t❛❜❧② ❝❤♦s❡♥ ♣❛r❛♠❡t❡rs ❤❛s ❢✉❧❧ ♠❡❛s✉r❡ ✭s❡❡ Pr♦♣♦s✐t✐♦♥ ✸✳✻✮✱ t❤❡❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✐s ✉s❡❢✉❧ ❢♦r ❝♦♠♣✉t❛t✐♦♥s ❝♦♥❝❡r♥✐♥❣ q✉❛♥t✐t❛t✐✈❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ ♥❡❛r❜② ♣♦✐♥ts✭s✐♥❝❡ ✐t ❛❧❧♦✇s t♦ t❤r♦✇ s❡ts ✇❤✐❝❤ ❛r❡ t❛✐❧s ♦❢ t❤❡ ❝♦♥✈❡r❣✐♥❣ s❡r✐❡s ❣✐✈❡♥ ❜② ❈♦r♦❧❧❛r② ✹✳✽✮✳ ❋r♦♠ ♥♦✇♦♥✱ ✐♥ t❤❡ r❡st ♦❢ t❤❡ ♣❛♣❡r ✇❡ ✇✐❧❧ ♦♥❧② ✉s❡ t❤❡ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ❛♥❞ ❤❡♥❝❡ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r ♦♥❧②■❊❚s ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❘❛t♥❡r ❉❈ ❢♦r ❛ r❛♥❣❡ ♦❢ ♣❛r❛♠❡t❡rs ✇❤✐❝❤ ✐♠♣❧② t❤❡ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥✳❇② ❈♦r♦❧❧❛r② ✹✳✾✱ t❤✐s s❡t ♦❢ ■❊❚s ❤❛s ❢✉❧❧ ♠❡❛s✉r❡ ❢♦r s♦♠❡ ❝❤♦✐❝❡ ♦❢ ❡①♣♦♥❡♥ts✳

❋r♦♠ ♥♦✇ ♦♥✱ ✐♥ ♦r❞❡r t♦ t♦ s✐♠♣❧✐❢② ♥♦t❛t✐♦♥✱ ✇❡ ✇✐❧❧ ✉s❡ ❡①♣♦♥❡♥ts (τ, ξ, η) ✭✐♥st❡❛❞ ♦❢ (τ ′, ξ′, η′✮✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥✳

✺✳ Pr♦♦❢ ♦❢ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt②

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ♣r♦✈❡ t❤❛t s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ■❊❚s ✉♥❞❡r ❢✉♥❝t✐♦♥s ✇✐t❤ ❧♦❣❛r✐t❤♠✐❝ ❛s②♠♠❡tr✐❝s✐♥❣✉❧❛r✐t✐❡s ❤❛✈❡ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ✇❤❡♥ t❤❡ ❜❛s❡ ■❊❚ s❛t✐s✜❡s t❤❡ ❘❛t♥❡r ❉❈ ✇✐t❤ ❛♥❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡ ♦❢ ♣❛r❛♠❡t❡rs ✭❚❤❡♦r❡♠ ✶✳✹✮✳ ■♥ ❙❡❝t✐♦♥ ✺✳✶✱ ✇❡ ✜rst ✉s❡ ❜❛❧❛♥❝❡❞ ❘❛✉③②✲❱❡❡❝❤t✐♠❡s t♦ s❤♦✇ t❤❛t ♦♥❡ ❝❛♥ ❝♦♥tr♦❧ t❤❡ ❞✐st❛♥❝❡ ♦❢ ♦r❜✐ts ♦❢ ♠♦st ♣♦✐♥ts ❢r♦♠ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ❡✐t❤❡r ✐♥t❤❡ ♣❛st ♦r ✐♥ t❤❡ ❢✉t✉r❡✳ ❚❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✹ ✐s t❤❡♥ ❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✺✳✷✳ ❋✐♥❛❧❧②✱ ✐♥ ❙❡❝t✐♦♥ ✺✳✸✇❡ ❞❡❞✉❝❡ t❤❡ ♦t❤❡r r❡s✉❧ts st❛t❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✳

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✸✶

✺✳✶✳ ❈♦♥tr♦❧ ♦❢ ❡✐t❤❡r ❜❛❝❦✇❛r❞ ♦r ❢♦r✇❛r❞ ♦r❜✐ts ❞✐st❛♥❝❡ ❢r♦♠ s✐♥❣✉❧❛r✐t✐❡s✳ ■♥ t❤✐s s❡❝t✐♦♥✇❡ s❤♦✇ t❤❛t ❜❛❧❛♥❝❡❞ ♣♦s✐t✐✈❡ t✐♠❡s ♦❢ t❤❡ ❘❛✉③② ❱❡❡❝❤ ✐♥❞✉❝t✐♦♥ ❛❧❧♦✇s ✉s t♦ ❝♦♥tr♦❧ t❤❡ ❞✐st❛♥❝❡♦❢ ♠♦st ♦r❜✐ts ❢r♦♠ t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ❡✐t❤❡r ❜❛❝❦✇❛r❞ ♦r ❢♦r✇❛r❞✱ ✐✳❡✳ ♣♦✐♥ts ✇❤♦ ❣❡t t♦♦ ❝❧♦s❡ t♦ ❛❞✐s❝♦♥t✐♥✉✐t② ✐♥ t❤❡ ❢✉t✉r❡✱ ❞♦ ♥♦t ❣❡t t♦♦ ❝❧♦s❡ ✐♥ t❤❡ ♣❛st ✭✇❤❡r❡ t♦♦ ❝❧♦s❡ ✐s q✉❛♥t✐✜❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥✺✳✶✮✳ ❚❤✐s ✇✐❧❧ ♣r♦✈✐❞❡ ❛ ❦❡② st❡♣ t♦ ♣r♦✈❡ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt②✱ s✐♥❝❡ ❛❝❝♦r❞✐♥❣ t♦ ✇❤❡t❤❡r❜❛❝❦✇❛r❞ ♦r ❢♦r✇❛r❞ ♦r❜✐ts st❛② ❢❛r ❢r♦♠ s✐♥❣✉❧❛r✐t✐❡s✱ ✇❡ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ✈❡r✐❢② t❤❡ ♣❛r❛❜♦❧✐❝ ❞✐✈❡r❣❡♥❝❡❡st✐♠❛t❡s ❡✐t❤❡r ✐♥ t❤❡ ❢✉t✉r❡ ♦❢ ✐♥ t❤❡ ♣❛st✳ ❚❤❡ ♠❛✐♥ ♣r♦♣♦s✐t✐♦♥ t❤❛t ✇❡ ♣r♦✈❡ ✐♥ t❤✐s s❡❝t✐♦♥ ✐sPr♦♣♦s✐t✐♦♥ ✺✳✶ st❛t❡❞ ❤❡r❡ ❜❡❧♦✇✳

▲❡t ✉s ✜rst r❡❝❛❧❧ t❤❛t Iα = [lα, rα) ❞❡♥♦t❡ t❤❡ ✐♥t❡r✈❛❧s ❡①❝❤❛♥❣❡❞ ❜② T = (λ, π)✳ ●✐✈❡♥ t✇♦ s❡tsE,F ⊂ [0, 1] ❧❡t ✉s ❞❡♥♦t❡ ❜② d(E,F ) t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ s❡ts✱ ❣✐✈❡♥ ❜②

✭✺✳✶✮ d(E,F ) = inf{|x− y|, x ∈ E, y ∈ F}.

Pr♦♣♦s✐t✐♦♥ ✺✳✶ ✭❇❛❝❦✇❛r❞ ♦r ❢♦r✇❛r❞ ❝♦♥tr♦❧✮✳ ▲❡t T ❜❡ ❛♥ ■❊❚ ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥❛♥❞ ❧❡t {nℓ}ℓ∈N ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ν✲❜❛❧❛♥❝❡❞ ✐♥❞✉❝t✐♦♥ t✐♠❡s ❢♦r T s✉❝❤ t❤❛t {nℓk}k∈N ✐s ❛ ♣♦s✐t✐✈❡

s❡q✉❡♥❝❡ ♦❢ t✐♠❡s ❢♦r s♦♠❡ ℓ ∈ N✳ ▲❡t qℓ ❞❡♥♦t❡ t❤❡ ♠❛①✐♠❛❧ ❤❡✐❣❤t ♦❢ t♦✇❡rs ♦❢ st❡♣ nℓ ✭s❡❡ ✭✸✳✷✮✮✳❚❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡r L ≥ 1✱ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ❜② L = ℓ(1+

[logd(2ν

2)]) ✭✇❤❡r❡ [·] ❞❡♥♦t❡s t❤❡ ✐♥t❡❣❡r

♣❛rt ❛♥❞ logd t❤❡ ❧♦❣❛r✐t❤♠ ✐♥ ❜❛s❡ d✮✱ ❛♥❞ c > 0 s✉❝❤ t❤❛t ❢♦r ❛♥② ε > 0✱ t❤❡r❡ ❡①✐sts ℓ′ = ℓ′(ε) ≥ 1s✉❝❤ t❤❛t ❢♦r ℓ ≥ ℓ′ ❛♥❞ x 6∈ [0, ε/8) ∪ (1− ε/8, 1)✱ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿

d({lα, rα : α ∈ A}, {T ix : 0 ≤ i < qℓ}) >c

qℓ+L,✭✺✳✷✮

d({lα, rα : α ∈ A}, {T ix : −qℓ ≤ i < 0}) >c

qℓ+L,✭✺✳✸✮

✇❤❡r❡ d(·, ·) ❞❡♥♦t❡s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ s❡ts ❞❡✜♥❡❞ ✐♥ ✭✺✳✶✮✳

❲❡ ✇✐❧❧ ✜rst st❛t❡ s♦♠❡ ❛✉①✐❧✐❛r② ❞❡✜♥✐t✐♦♥s ❛♥❞ t✇♦ ▲❡♠♠❛s ❛♥❞ t❤❡♥ ✉s❡ t❤❡♠ t♦ ♣r♦✈❡ Pr♦♣♦s✐t✐♦♥✺✳✶✳ ●✐✈❡♥ T = (λ, π)✱ r❡♠❛r❦ t❤❛t t❤❡ ✐♥t❡r✈❛❧ ❡♥❞♣♦✐♥ts 0 ❛♥❞ 1 ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s 0 = lα1,t ❛♥❞1 = rαd,t ✇❤❡r❡ α1,t ❛♥❞ αd,t ❛r❡ ❞❡✜♥❡❞ r❡s♣❡❝t✐✈❡❧② t♦ ❜❡ t❤❡ ❧❡tt❡rs ✐♥ A s✉❝❤ t❤❛t πt(α1,t) = 1❛♥❞ πt(αd,t) = d✱ s♦ t❤❛t Iα1,t ❛♥❞ Iαd,t ❛r❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✜rst ❛♥❞ ❧❛st ✐♥t❡r✈❛❧ ❜❡❢♦r❡ t❤❡ ❡①❝❤❛♥❣❡✳▼♦r❡♦✈❡r✱ ✐❢ α1,b✱ αd,b ❛r❡ s✉❝❤ t❤❛t πb(α1,b) = 1 ❛♥❞ πb(αd,b) = d✱ Iα1,b

❛♥❞ Iαd,b ❛r❡ s✉❝❤ t❤❛t t❤❡✐r✐♠❛❣❡ ✉♥❞❡r T ❛r❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✜rst ❛♥❞ ❧❛st ✐♥t❡r✈❛❧ ❛❢t❡r t❤❡ ❡①❝❤❛♥❣❡✳

❘❡♠❛r❦ ✺✳✷✳ ❲❡ r❡♠❛r❦ t❤❛t ❢♦r ❛♥② α ∈ A s✉❝❤ t❤❛t α 6= α1,t✱ lα = rβ ✇❤❡r❡ β ✐s s✉❝❤ t❤❛tπt(β) = πt(α)− 1✱ s♦ t❤❛t Iβ ✐s t❤❡ ✐♥t❡r✈❛❧ ✇❤✐❝❤ ♣r❡❝❡❡❞s Iα ❜❡❢♦r❡ t❤❡ ❡①❝❤❛♥❣❡✳ ❙✐♠✐❧❛r❧②✱ ❢♦r ❛♥②α ∈ A s✉❝❤ t❤❛t α 6= αd,t✱ rα = lβ ✇❤❡r❡ β ✐s s✉❝❤ t❤❛t πt(β) = πt(α) + 1✱ s♦ t❤❛t Iβ ✐s t❤❡ ✐♥t❡r✈❛❧✇❤✐❝❤ ❢♦❧❧♦✇s Iα ❜❡❢♦r❡ t❤❡ ❡①❝❤❛♥❣❡✳

●✐✈❡♥ ❛♥ ■❊❚ T (n) = (λ(n), π(n)) ✐♥ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ♦r❜✐t ♦❢ T ✱ ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② I(n)α =

[l(n)α,t , r

(n)α,t

)

❢♦r α ∈ A t❤❡ ✐♥t❡r✈❛❧s ❡①❝❤❛♥❣❡❞ ❜② T (n) ❛♥❞ ✇❡ ✇✐❧❧ ❞❡♥♦t❡ t❤❡✐r ✐♠❛❣❡s ✉♥❞❡r T (n) ❜②[l(n)α,b , r

(n)α,b

)✳

❊①♣❧✐❝✐t❡❧②✱ t❤❡ ❡♥❞♣♦✐♥ts ❛r❡ ❣✐✈❡♥ ❜②

l(n)α,t :=

∑

π(n)t (β)<π

(n)t (α)

λ(n)β , r

(n)α,t :=

∑

π(n)t (β)≤π

(n)t (α)

λ(n)β ;✭✺✳✹✮

l(n)α,b :=

∑

π(n)b

(β)<π(n)b

(α)

λ(n)β , r

(n)α,b :=

∑

π(n)b

(β)≤π(n)b

(α)

λ(n)β .✭✺✳✺✮

❲❡ ✇✐❧❧ ❛❧s♦ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ α(n)1,t ✱ α

(n)d,t ✱ α

(n)1,b ✱ α

(n)d,b ❢♦r t❤❡ ❧❡tt❡rs s✉❝❤ t❤❛t

✭✺✳✻✮ π(n)t (α

(n)1,t ) = 1, π

(n)t (α

(n)d,t ) = d, π

(n)b (α

(n)1,b ) = 1, π

(n)b (α

(n)d,b ) = d.

❆ ❝r✉❝✐❛❧ st❡♣ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ▲❡♠♠❛✱ ✇❤✐❝❤ ✐s ❛ s♠❛❧❧ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ❈♦r♦❧❧❛r② ❈✳✷ ✭s❡❡ ❛❧s♦▲❡♠♠❛ ❈✳✶✮ ✐♥ t❤❡ ❆♣♣❡♥❞✐① ♦❢ ❬✶✹❪✳ ❋♦r ❝♦♠♣❧❡t❡♥❡ss✱ ✇❡ ✐♥❝❧✉❞❡ ✐ts s❤♦rt ♣r♦♦❢ ✐♥ t❤❡ ❆♣♣❡♥❞✐① ❆✳✷✳

✸✷ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

▲❡♠♠❛ ✺✳✸ ✭s❡❡ ❈♦r♦❧❧❛r② ❈✳✷ ✐♥ ❬✶✹❪ ❛♥❞ ❆♣♣❡♥❞✐① ❆✳✷✮✳ ▲❡t T ✱ {nℓ}ℓ∈N ❛♥❞ qℓ ❜❡ ❛s ✐♥ Pr♦♣♦s✐t✐♦♥

✺✳✶ ❛♥❞ ❧❡t α(nℓ)1,b ❛♥❞ α

(nℓ)d,t ❜❡ ❛s ✐♥ ❛♥❞ ✭✺✳✻✮✳ ❚❤❡♥

min{

|l(nℓ)α,t − l

(nℓ)β,b |, α ∈ A, β ∈ A\{α

(nℓ)1,b }

}≥

1

νλ(nℓ+ℓ),✭✺✳✼✮

min{

|r(nℓ)α,t − r

(nℓ)β,b |, α ∈ A\{α

(nℓ)d,t }, β ∈ A

}≥

1

νλ(nℓ+ℓ).✭✺✳✽✮

❯s✐♥❣ ▲❡♠♠❛ ✺✳✸✱ ✇❡ ❝❛♥ t❤❡♥ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳

▲❡♠♠❛ ✺✳✹✳ ❙✉♣♣♦s❡ t❤❛t {nℓ}ℓ∈N ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ν✲❜❛❧❛♥❝❡❞ ❛❝❝❡❧❡r❛t✐♦♥ t✐♠❡s ❢♦r T s✉❝❤ t❤❛t{nℓk}k∈N ✐s ♣♦s✐t✐✈❡ ❢♦r s♦♠❡ ℓ ∈ N✳ ❙❡t L : ℓ(1 +

[logd(2ν

2)])✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ℓ s✉✣❝✐❡♥t❧② ❧❛r❣❡✱

⋃

0≤i<2qℓ

T i([lα, lα +

1

3νqℓ+L

])∩ {lβ , rβ : β ∈ A} = ∅, ❢♦r ❛❧❧ lα s✳t✳ T (lα) 6= 0, ✐.e. s.t. α 6= α1

b ;

✭✺✳✾✮

⋃

0≤i<2qℓ

T i([rα −

1

3νqℓ+L, rα

))∩ {lβ , rβ : β ∈ A} = ∅, ❢♦r ❛❧❧ rα 6= 1, ✐.e. s.t. α 6= αdt .

✭✺✳✶✵✮

❲❡ r❡❝❛❧❧ t❤❛t t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ 0, 1 ❢♦❧❧♦✇s ❢r♦♠ ❘❡♠❛r❦ ✺✳✷✳

Pr♦♦❢✳ ❘❡❝❛❧❧ ✜rst s♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ t♦✇❡rs Z(nℓ)α ✱ α ∈ A✱ ℓ ≥ 1 ✐♥ t❤❡ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥

✭❞❡✜♥❡❞ ❜② ✭✷✳✹✮ ✐♥ ❙❡❝t✐♦♥ ✷✳✺✮✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ Rnℓ(T ) ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✜rst

✈✐s✐t ✈✐❛ T−1 t♦ I(nℓ) ♦❢ t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ❢♦r T ✳ ❚❤❡r❡❢♦r❡ ✭s❡❡ ❋✐❣✉r❡ ✸✭❛✮✮✿

✭❛✮ ❢♦r ❡❛❝❤ α ∈ A✱ ✐♥ t❤❡ t♦✇❡r Z(nℓ)α t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ✐♥t❡r✈❛❧ T iαI

(nℓ)α ✱ ❢♦r 0 ≤ iα ≤ h

(nℓ)α − 1✱

✇❤♦s❡ ❧❡❢t ❡♥❞♣♦✐♥t ❜❡❧♦♥❣s t♦ {lβ : β ∈ A}❀ ♠♦r❡ ♣r❡❝✐s❡❧②✱ t❤✐s ❧❡❢t ❡♥❞♣♦✐♥t ✐s lα❀

✭❜✮ ❢♦r ❡❛❝❤ α ∈ A✱ ✇✐t❤ t❤❡ ❡①❝❧✉s✐♦♥ ♦❢ α(nℓ+ℓ0 )

1,t ✭s❡❡ ✭✺✳✻✮✮✱ t❤❡r❡ ✐s ♦♥❡ ✐♥t❡r✈❛❧ ✐♥ t❤❡ t♦✇❡r Z(nℓ)α ✱

t❤❛t ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② T jαI(nℓ)α ✱ 0 ≤ jα ≤ h

(nℓ)α − 1✱ ✇❤♦s❡ r✐❣❤t ❡♥❞♣♦✐♥t ❜❡❧♦♥❣s t♦ {rβ : β ∈ A}❀

♠♦r❡ ♣r❡❝✐s❡❧②✱ t❤✐s r✐❣❤t ❡♥❞♣♦✐♥t ✐s rα′ ✱ ✇❤❡r❡ α′ ✐s s✉❝❤ t❤❛t rα′ = lα ✭s❡❡ ❘❡♠❛r❦ ✺✳✷✱ ✇❤✐❝❤❣✐✈❡s t❤❛t ❡①♣❧✐❝✐t❧② α′ + π−1

t (πt(α)− 1)✮❀✭❝✮ t❤❡ ❡♥❞♣♦✐♥t 1 ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✐♥t❡r✈❛❧ ✭✇❤✐❝❤ ✐s ❛❧s♦ ❡q✉❛❧ t♦ rαd,t ✱ s❡❡ ❥✉st ❜❡❢♦r❡ t❤❡ ❘❡♠❛r❦ ✺✳✷✮✱

✐s t❤❡ ✐♠❛❣❡ ♦❢ rαd,b ❛♥❞ ❤❡♥❝❡✱ ❜② ✭❜✮✱ ✐s t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♦❢ ❛ ✢♦♦r T iI(nℓ)β ♦❢ Z

(nℓ)β ✱ ❢♦r s♦♠❡

0 ≤ i ≤ h(nℓ)β − 1✱ t❤❡♥ 1 = rαt

d✐s t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♦❢ t❤❡ ✢♦♦r T i+1I

(nℓ)β ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ Z

(nℓ)β

❝♦♥t❛✐♥s t✇♦ ❡❧❡♠❡♥ts ♦❢ {rα : α ∈ A} ♦♥❡ ❛❜♦✈❡ t❤❡ ♦t❤❡r ✭s❡❡ ❋✐❣✉r❡ ✸✭❛✮✮❀

✭❞✮ t❤❡ ♣♦✐♥t T−1(0) ✐s t❤❡ ❧❡❢t ❡♥❞♣♦✐♥t ♦❢ t❤❡ t♦♣ ✢♦♦r ♦❢ t❤❡ t♦✇❡r Z(nℓ)

α(n)1,b

✳

▲❡t ✉s ❝❤♦♦s❡ ❛ st❡♣ nℓ+ℓ0 ✇❤♦s❡ t♦✇❡rs ❛r❡ ❛❧❧ t❛❧❧❡r t❤❛♥ t✇✐❝❡ t❤❡ s❤♦rt❡st t♦✇❡r ♦❢ st❡♣ nℓ✳ ❲❡❝❧❛✐♠ t❤❛t ✇❡ ❤❛✈❡ t❤❛t

✭✺✳✶✶✮ minα∈A

h(nℓ+ℓ0 )α ≥ 2qℓ + 2max

α∈Ah(nℓ+ℓ0 )α , ✇❤❡r❡ ℓ0 +

[logd(2ν

2)]ℓ.

■♥❞❡❡❞✱ s✐♥❝❡ (nℓ)ℓ ✐s ❜② ❛ss✉♠♣t✐♦♥ ν✲❜❛❧❛♥❝❡❞ ✭s❡❡ ❘❡♠❛r❦ ✷✳✶✶✮✱ (nkℓ)k ✐s ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ ♦❢✐♥❞✉❝t✐♦♥ t✐♠❡s ❛♥❞ ❜② ♣♦s✐t✐✈✐t② ✭s❡❡ ❘❡♠❛r❦ ✷✳✶✵✮ ✐t ❢♦❧❧♦✇s t❤❛t ❢♦r ❛♥② k ≥ 1✱ ✇❡ ❤❛✈❡

minα∈A

h(nℓ+kℓ)

α ≥1

νλ(nℓ+kℓ)≥

dk

νλ(nℓ)≥dk

ν2maxα∈A

h(nℓ)α ,

s♦ ❢♦r k = ℓ0/ℓ✱ ✇❡ ❤❛✈❡ t❤❛t dk > 2ν2 ❛♥❞ ❤❡♥❝❡ ❣❡t ✭✺✳✶✶✮ ❛s ❞❡s✐r❡❞✳❙❡t L + ℓ0 + ℓ = ℓ(1 +

[logd(2ν

2)])✱ ✇❤❡r❡ ν ✐s t❤❡ ❜❛❧❛♥❝❡ ❝♦♥st❛♥t ❛♥❞ ℓ s✉❝❤ t❤❛t (nkℓ)k ✐s ❛

♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ ♦❢ ✐♥❞✉❝t✐♦♥ t✐♠❡s✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉❧❧ ❤❡✐❣❤t s✉❜t♦✇❡rs ♦❢ t❤❡ t♦✇❡rs Z(nℓ+ℓ0 )α

✭s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✸✭❜✮ ✐♥ ❧✐❣❤t❡r s❤❛❞❡✮✱ ✇❤✐❝❤ ❤❛✈❡ ✇✐❞t❤s 1/(3νqℓ+L) ❛♥❞ ✇❤♦s❡ ❜❛s❡s ❝♦♥t❛✐♥ ❛s

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✸✸

✭❛✮ ❚♦✇❡rs ❛t ❧❡✈❡❧ nℓ✳ ❍❡r❡✱ d = 4 ❛♥❞πb(α

′

3) = d✳✭❜✮ ❙✉❜t♦✇❡rs ✐♥ ✭✺✳✶✷✮✱ ✭✺✳✶✸✮ ✭❧✐❣❤t❡r s❤❛❞❡✮

❛♥❞ ✭✺✳✶✹✮✱ ✭✺✳✶✺✮ ✭❞❛r❦❡r s❤❛❞❡✮✳

❋✐❣✉r❡ ✸✳ ❚❤❡ s✉❜t♦✇❡rs ✉s❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✺✳✹✿ t❤❡ t♦✇❡rs ✐♥ ✭✺✳✶✷✮✱ ✭✺✳✶✸✮ ❛r❡ ✐♥❧✐❣❤t❡r s❤❛❞❡✱ t❤❡ ♦♥❡s ❣✐✈❡♥ ❜② ✭✺✳✶✹✮✱ ✭✺✳✶✺✮ ❛r❡ ✐♥ ❞❛r❦❡r s❤❛❞❡✳

❡♥❞♣♦✐♥ts t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ T (nℓ+ℓ0 )✿

⋃

0≤i<h(nℓ+ℓ0

)

α

T i([lnℓ+ℓ0 ,tα , l

nℓ+ℓ0 ,tα +

1

3νqℓ+L

]), α ∈ A;✭✺✳✶✷✮

⋃

0≤i<h(nℓ+ℓ0

)

α

T i([rnℓ+ℓ0 ,tα −

1

3νqℓ+L, r

nℓ+ℓ0 ,tα

)), α ∈ A\{α

(nℓ+ℓ0 )

d,t }.✭✺✳✶✸✮

❈♦♥s✐❞❡r ❛❧s♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉❧❧ ❤❡✐❣❤t s✉❜t♦✇❡rs ♦❢ t❤❡ s❛♠❡ ✇✐❞t❤ ✭s❤♦✇♥ ✐♥ ❞❛r❦❡r s❤❛❞❡ ✐♥ ❋✐❣✲

✉r❡ ✸✭❜✮✮✱ ✇❤♦s❡ ❜❛s❡s ❝♦♥t❛✐♥ ❛s ❡♥❞♣♦✐♥ts t❤❡ ✐♠❛❣❡s ♦❢ t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ T (nℓ+ℓ0 )✿

⋃

0≤i<h(nℓ+ℓ0

)

β(α)

T i([lnℓ+ℓ0 ,bα , l

nℓ+ℓ0 ,bα +

1

3νqℓ+L

]), α ∈ A\{α

(nℓ+ℓ0 )

1,b }, β(α) s✳t✳ lnℓ+ℓ0 ,tα ∈ I

(nℓ+ℓ0 )

β(α) ;

✭✺✳✶✹✮

⋃

0≤i<h(nℓ+ℓ0

)

β(α)

T i([rnℓ+ℓ0 ,bα −

1

3νqℓ+L, r

nℓ+ℓ0 ,bα

)), α ∈ A, β(α) s✳t✳ r

nℓ+ℓ0 ,tα ∈ I

(nℓ+ℓ0 )

β(α) .

✭✺✳✶✺✮

❚❤❡ ❦❡② r❡♠❛r❦ t❤❛t ❢♦❧❧♦✇s ❢r♦♠ ▲❡♠♠❛ ✺✳✸ ✐s t❤❛t t❤❡ s✉❜t♦✇❡rs ✐♥ ✭✺✳✶✷✮✱ ✭✺✳✶✸✮✱ ✭✺✳✶✹✮ ❛♥❞ ✭✺✳✶✺✮❛r❡ ❛❧❧ ♣❛✐r✇✐s❡ ❞✐s❥♦✐♥t✱ s✐♥❝❡ t❤❡✐r ✇✐❞t❤ 1/3νqℓ+L ✐s ❧❡ss t❤❛♥ ❤❛❧❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❡♥❞♣♦✐♥ts✐♥ t❤❡✐r ❜❛s❡ ✢♦♦rs ✭✇❤✐❝❤ ✐s 1/(νqℓ+L) ❜② ▲❡♠♠❛ ✺✳✸✮✳ ❚❤✉s✱ s✐♥❝❡ ❜② ♣r♦♣❡rt✐❡s ✭❛✮✱ ✭❜✮ ❛♥❞ ✭❝✮r❡❝❛❧❧❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ {lα, rα : α ∈ A} ❛r❡ ❡♥❞♣♦✐♥ts ♦❢ ✢♦♦rs ♦❢ t❤❡ s✉❜t♦✇❡rs ✐♥✭✺✳✶✷✮ ❛♥❞ ✭✺✳✶✸✮✱ t❤❡ ❝❧♦s✉r❡ ♦❢ t❤❡ s✉❜t♦✇❡rs ✐♥ ✭✺✳✶✹✮ ❛♥❞ ✭✺✳✶✺✮ ❞♦❡s ♥♦t ✐♥t❡rs❡❝t {lα, rα : α ∈ A}✳

▲❡t ✉s ♥♦✇ ♣r♦✈❡ ✭✺✳✾✮✳ ❚❛❦❡ ❛♥② α 6= α1b ✱ s♦ t❤❛t T (lα) 6= 0✳ ❇② Pr♦♣❡rt② ✭❛✮ r❡❝❛❧❧❡❞ ❛t t❤❡

❜❡❣✐♥♥✐♥❣✱[lα, lα + 1

(3νqℓ+L)

]✐s ❛ ✢♦♦r ♦❢ ♦♥❡ ♦❢ t❤❡ s✉❜t♦✇❡r ✐♥ ✭✺✳✶✷✮✱ t❤❡ ♦♥❡ ✐♥❞❡①❡❞ ❜② t❤❡ s❛♠❡

α✳ ▼♦r❡♦✈❡r✱ ❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ T (lα) 6= 0 ❛♥❞ ♣r♦♣❡rt② ✭❞✮✱ ✐t ❢♦❧❧♦✇s t❤❛t π(nℓ+ℓ0 )

b (α) 6= 1 ❛♥❞t❤❛t t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❧❛st ✢♦♦r ♦❢ t❤❡ α s✉❜t♦✇❡r ✐s t❤❡ ❜❛s❡ ♦❢ ❛ s✉❜t♦✇❡r ✐♥ ✭✺✳✶✹✮✳ ❚❤✉s✱ s✐♥❝❡ ❜②

✭✺✳✶✶✮ 2qℓ ≤ minα∈A h(nℓ+ℓ0 )α ✱ t❤❡ ✐♠❛❣❡s ♦❢ t❤❡ ✐♥t❡r✈❛❧

[lα, lα + 1

(3νqℓ+L)

]✉♥❞❡r T i ❢♦r 0 ≤ i < 2qℓ

✸✹ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

❛r❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ α s✉❜t♦✇❡r ✐♥ ✭✺✳✶✷✮ ❛♥❞ ♦❢ ❛ s✉❜t♦✇❡r ✐♥ ✭✺✳✶✹✮ ✭♠♦r❡ ♣r❡❝✐s❡❧②✱

t❤❡ s✉❜t♦✇❡r ✇❤✐❝❤ ❤❛s ❛s ❡♥❞♣♦✐♥t l(nℓ+ℓ0 )α,n ✱ ✇❤✐❝❤ ✐s t❤❡ ✐♠❛❣❡ ♦❢ l

(nℓ+ℓ0 )α,t ✉♥❞❡r T (nℓ+ℓ0 )✮✳ ❚❤✉s✱ t♦

♣r♦✈❡ ✭✺✳✾✮ ❤♦❧❞s ✐t ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ t❤❛t ♥❡✐t❤❡r ♦❢ t❤❡s❡ t✇♦ s✉❜t♦✇❡rs ❝♦♥t❛✐♥ ♦t❤❡r ❡❧❡♠❡♥ts ♦❢{lα, rα : α ∈ A}✳ ❚❤✐s ✐s t❤❡ ❝❛s❡ s✐♥❝❡ ❜② t❤❡ ♣r♦♣❡rt✐❡s ✭❛✮✱ ✭❜✮ ❛♥❞ ✭❝✮✱ t❤❡ α s✉❜t♦✇❡r ✐♥ ✭✺✳✶✷✮ ❞♦❡s♥♦t ❝♦♥t❛✐♥ ❛♥② ♦t❤❡r ❡❧❡♠❡♥t ♦❢ {lα, rα : α ∈ A} ❛♣❛rt lα ❛s ❧❡❢t ❡♥❞♣♦✐♥t ❛♥❞✱ ❛s r❡♠❛r❦❡❞ ❛❜♦✈❡✱ t❤❡❝❧♦s✉r❡ ♦❢ t❤❡ s✉❜t♦✇❡rs ✐♥ ✭✺✳✶✹✮ ❞♦❡s ♥♦t ✐♥t❡rs❡❝t {lα, rα : α ∈ A}✳ ❚❤✉s✱ ✭✺✳✾✮ ❤♦❧❞s✳

▲❡t ✉s ♥♦✇ ♣r♦✈❡ ✭✺✳✶✵✮✳ ❚❛❦❡ ❛♥② α 6= αtd✱ s♦ t❤❛t rα 6= 1✳ ❇② Pr♦♣❡rt② ✭❜✮ r❡❝❛❧❧❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣✱[rα − 1

(3νqℓ+L), rα

)✐s ❛ ✢♦♦r ♦❢ ♦♥❡ ♦❢ t❤❡ s✉❜t♦✇❡rs ✭✺✳✶✷✮✳ ❇② ❛❧❧ ♣r♦♣❡rt✐❡s ✭❛✮✲✭❞✮✱ s✐♥❝❡ rα 6= 1✱ t❤❡

✢♦♦rs ♦❢ t❤✐s s✉❜t♦✇❡r ❛❜♦✈❡ t❤❡ ✢♦♦r ✇❤✐❝❤ ❝♦♥t❛✐♥s rα ❛s r✐❣❤t ❡♥❞♣♦✐♥t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛♥② ♦t❤❡r❞✐s❝♦♥t✐♥✉✐t② ✐♥ {lα, rα : α ∈ A} ✐♥ t❤❡✐r ❝❧♦s✉r❡✳ ❆s ❜❡❢♦r❡✱ ❜② ✭✺✳✶✶✮ t❤❡ ✐♠❛❣❡s ♦❢ t❤✐s ✐♥t❡r✈❛❧ ✉♥❞❡rT i ❢♦r 0 ≤ i < 2qℓ ❛r❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ♦r✐❣✐♥❛❧ s✉❜t♦✇❡r ♦❢ ✭✺✳✶✹✮ ❛♥❞ ♦♥❡ ♦❢ t❤❡ s✉❜t♦✇❡r ♦❢ ✭✺✳✶✺✮

✭t❤❡ ♦♥❡ ✇❤✐❝❤ ❤❛s ❛s ❡♥❞♣♦✐♥t rnℓ+ℓ0 ,b

α′ ✱ ✇❤✐❝❤ ✐s t❤❡ ✐♠❛❣❡ ♦❢ rnℓ+ℓ0 ,t

α′ ✉♥❞❡r T (nℓ+ℓ0 )✮✳ ❚❤✉s✱ s✐♥❝❡ t❤❡❝❧♦s✉r❡ ♦❢ t❤❡ s✉❜t♦✇❡rs ✐♥ ✭✺✳✶✺✮ ❛❧s♦ ❞♦❡s ♥♦t ✐♥t❡rs❡❝t {lα, rα : α ∈ A}✱ ✭✺✳✶✵✮ ❤♦❧❞s✳ �

Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✺✳✶✳ ▲❡t L = ℓ(1+[logd(2ν

2)]) ❛♥❞ s❡t c + 1/(6ν)✳ ●✐✈❡♥ ε > 0✱ ❧❡t ℓ′ = ℓ′(ε) ≥ 1

s✉❝❤ t❤❛t λ(nℓ′ ) ≤ ε/8✳ ❋✐① ℓ ≥ ℓ′✳ ❆ss✉♠❡ t❤❛t ♥❡✐t❤❡r ✭✺✳✷✮ ♥♦r ✭✺✳✸✮ ❤♦❧❞✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts t✇♦❝♦♥t✐♥✉✐t② ✐♥t❡r✈❛❧s ❡♥❞♣♦✐♥ts e1, e2 ∈ {lα, rα : α ∈ A} ❛♥❞ i1, i2 ∈ N ✇✐t❤

✭✺✳✶✻✮ − qℓ ≤ −i1 < 0, 0 ≤ i2 < qℓ,

s✉❝❤ t❤❛t

✭✺✳✶✼✮ |T−i1(x)− e1| <c

qℓ+L, |T i2(x)− e2| <

c

qℓ+L

✭s❡❡ ❋✐❣✉r❡ ✹ ❢♦r ❛ s❝❤❡♠❛t✐❝ ♣✐❝t✉r❡✮✳ ❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t i1, i2 ❛r❡ t❤❡s♠❛❧❧❡st ♥❛t✉r❛❧ ♥✉♠❜❡rs ✇❤✐❝❤ s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✳ ▲❡t ✉s ❝♦♥s✐❞❡r ✜rst t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ e1 ≤T−(i1+i2)e2 ✭t✇♦ s✉❝❤ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ ♣♦✐♥ts ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✭❛✮ ❛♥❞ ✹✭❜✮✮✳ ❇② ❘❡♠❛r❦ ✺✳✷✱ ✇❡❝❛♥ ❛ss✉♠❡ t❤❛t e1 = lα ❢♦r s♦♠❡ α ∈ A✳ ▲❡t ✉s s❤♦✇ t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t x /∈ [0, ε/8) ❣✉❛r❛♥t❡❡st❤❛t α 6= α1,b✳ ■♥❞❡❡❞✱ ✐❢ α = α1,b✱ t❤❡♥ T (lα) = 0✳ ■♥ t❤✐s ❝❛s❡✱ s✐♥❝❡ 0 ❛❧s♦ ❜❡❧♦♥❣s t♦ {lα, rα : α ∈ A}✱✇❡ ♠✉st ❤❛✈❡ −i1 = −1, i2 = 0 ❛♥❞ e2 = 0✳ ❚❤✉s✱ |x| = |T 0x − 0| ≤ c/(qℓ+L)✱ ✇❤✐❝❤✱ s✐♥❝❡ c < 1 ❛♥❞ℓ+L ≥ ℓ′✱ ❜② ❘❡♠❛r❦ ✷✳✶✶ ❣✐✈❡s |x| ≤ λℓ′ ✳ ❚❤✐s✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ℓ′✱ ✐♠♣❧✐❡s t❤❛t x ∈ [0, ε/8)✱ ✇❤✐❝❤ ✇❡❛r❡ ❡①❝❧✉❞✐♥❣ ❜② ❛ss✉♠♣t✐♦♥✳

✭❛✮ ✭❜✮ ✭❝✮ ✭❞✮

❋✐❣✉r❡ ✹✳ ✐♥ ✭❛✮ ❛♥❞ ✭❜✮ e1 ≤ T−(i1+i2)e2 ❛♥❞ e1 = lα❀ ✐♥ ✭❝✮ ❛♥❞ ✭❞✮ e1 ≥ T−(i1+i2)e2 ❛♥❞ e1 = rα✳

❲❡ r❡♠❛r❦ t❤❛t ❜② t❤❡ ❝❤♦✐❝❡ ♦❢ i1, i2✱ Tj ❛❝ts ❝♦♥t✐♥✉♦✉s❧② ♦♥ [lα, lα + 1/(3νqℓ+L)] ❢♦r ❛♥② 0 ≤ j ≤

i1+ i2 ❛♥❞ ❤❡♥❝❡ ✐t ✐s ❛♥ ✐s♦♠❡tr②✳ ❚❤✉s✱ ✉s✐♥❣ t❤❛t T i1+i2 ✐s ❛♥ ✐s♦♠❡tr② ❛♥❞ ✭✺✳✶✼✮ t✇✐❝❡ ❛♥❞ r❡❝❛❧❧✐♥❣t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ c✱ ✇❡ ❤❛✈❡ t❤❛t ✭s❡❡ ❋✐❣✉r❡ ✹✭❛✮ ❢♦r r❡❢❡r❡♥❝❡✮✿

|e2 − T i1+i2(e1)| = |e2 − T i2(x)|+ |T i2(x)− T i1+i2(e1)| ≤c

qℓ+L+ |T−i1(x)− e1| <

2c

qℓ+L=

1

3νqℓ+L.

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✸✺

❚❤✉s✱ ❜② ✭✺✳✶✻✮✱ t❤✐s s❤♦✇s t❤❛t

e2 ∈ T i1+i2([lα, lα +

1

3νqℓ+L

]), ✇❤❡r❡ 0 ≤ i1 + i2 < 2qℓ,

✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ✭✺✳✾✮ ✐♥ ▲❡♠♠❛ ✺✳✹✳

❙✐♠✐❧❛r❧②✱ ✐❢ e1 ≥ T−(i1+i2)e2 ✭t✇♦ s✉❝❤ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ ♣♦✐♥ts ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✭❝✮ ❛♥❞ ✹✭❞✮✮✱❜② ❘❡♠❛r❦ ✺✳✷✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t e1 = rα ❢♦r s♦♠❡ α ∈ A✳ ▼♦r❡♦✈❡r✱ t❤❡ ❛ss✉♠♣t✐♦♥ x /∈ (1− ε/8, 1)❣✉❛r❛♥t❡❡s t❤❛t α 6= αt,d✱ s✐♥❝❡ ♦t❤❡r✇✐s❡✱ s✐♥❝❡ rαt,d = 1 ❛♥❞ T (rαt,d,t) = rαt,d ✱ ✇❡ ♠✉st ❤❛✈❡ e2 = rαt,d ✱

e1 = rαt,d ❛♥❞ ❤❡♥❝❡ i1 = 0, i2 = −1✳ ❚❤✉s✱ r❡❛s♦♥✐♥❣ ❛s ❜❡❢♦r❡ t❤✐s ②✐❡❧❞s t❤❛t |x−1| = |T 0x− rαt,d | ≤1/(3νqℓ+L) ≤ λℓ′ ≤ ε/8 ❛♥❞ ❤❡♥❝❡ x ∈ (1− ε/8, 1)✱ ✇❤✐❝❤ ✇❡ ❛r❡ ❡①❝❧✉❞✐♥❣✳ ❘❡❛s♦♥✐♥❣ ✐♥ ❛ s✐♠✐❧❛r ✇❛②❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❛s❡✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

e2 ∈ T i1+i2([rα −

1

3νqℓ+L, rα

)), ✇❤❡r❡ 0 ≤ i1 + i2 < 2qℓ,

✇❤✐❝❤ t❤✐s t✐♠❡s ❝♦♥tr❛❞✐❝ts ✭✺✳✶✵✮ ✐♥ ▲❡♠♠❛ ✺✳✹✳�

✺✳✷✳ Pr♦♦❢ t❤❛t t❤❡ ❘❛t♥❡r ❉❈ ✐♠♣❧✐❡s t❤❡ ❙❘✲♣r♦♣❡rt②✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ♣r♦✈❡ t❤❛t ✐❢ ❛♥■❊❚ ✐s s✉❝❤ t❤❛t t❤❡ ❘❛t♥❡r ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ❤♦❧❞s✱ ♦♥❡ ❝❛♥ ♣r♦✈❡ t❤❡ ❙❘✲♣r♦♣❡rt② ❢♦r s♣❡❝✐❛❧✢♦✇s ♦✈❡r T ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ✇✐❧❧ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳

Pr♦♣♦s✐t✐♦♥ ✺✳✺✳ ▲❡t T : I → I ❜❡ ❛♥ ■❊❚ ❛♥❞ f : I → R+ ❛ r♦♦❢ ❢✉♥❝t✐♦♥ f ∈ AsymLogSing(T ) ✭s❡❡

❉❡✜♥✐t✐♦♥ ✷✳✶✮✳ ■❢ T s❛t✐s✜❡s t❤❡ ❘❛t♥❡r ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✇✐t❤ ❡①♣♦♥❡♥ts (τ, η, ξ) s✉❝❤ t❤❛t

✭✺✳✶✽✮ τ ∈ (1,16

15), τ ′ ∈ (

15

16, 1), η ∈ (3/4, 2τ ′ − τ), ξ ∈ (max(99/100, τ ′η), τ ′).

t❤❡ s♣❡❝✐❛❧ ✢♦✇ (ϕt)t∈R ♦✈❡r T ❛♥❞ ✉♥❞❡r f ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✳

❲❡ ✇✐❧❧ ❛ss✉♠❡ ❢♦r t❤❡ r❡st ♦❢ t❤❡ s❡❝t✐♦♥ t❤❛t t❤❡ ♣❛r❛♠❡t❡rs τ, τ ′, ξ, η ❛r❡ ❝❤♦s❡♥ ❛s ✐♥ ✭✺✳✶✽✮✱ t❤❛tT s❛t✐s✜❡s t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ Pr♦♣♦s✐t✐♦♥ ✺✳✺ ❛♥❞ t❤❛t f ∈ AsymLogSing(T )✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② C−✱C+ t❤❡ ❝♦♥st❛♥ts ✐♥ t❤❡ ❉❡✜♥✐t✐♦♥ ✷✳✶ ♦❢ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❛ss✉♠❡✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② t❤❛t C− − C+ > 0✳ ▲❡t ✉s ✜rst ❣✐✈❡ ❛♥ ♦✉t❧✐♥❡ ♦❢ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ♣r♦♦❢✳

❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢✳ ❲❡ ✇✐❧❧ ♣r♦✈❡ t❤❡ ❙❘✲♣r♦♣❡rt② ✉s✐♥❣ ▲❡♠♠❛ ✹✳✷ ♦♥ ❇✐r❦❤♦✛ s✉♠s t♦ ✈❡r✐❢②✐♥❣t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ❈♦♥s✐❞❡r x < y ❝❧♦s❡✳ ❚♦ ✉s❡ ▲❡♠♠❛ ✹✳✷ ✇❡ ♥❡❡❞ t✇♦ ✈❡r✐❢② t❤❛tt❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♣r♦♣❡rt✐❡s ❤♦❧❞✿

✭✐✮ t❤❡r❡ ✐s ♥♦ ❞✐s❝♦♥t✐♥✉✐t② ✐♥ [T ix, T iy] ❢♦r i ∈ [M,M + L]❀✭✐✐✮ ✇❡ ❤❛✈❡ ❣♦♦❞ ❝♦♥tr♦❧ ♦❢ Sn(f

′)(θ) ❢♦r n ∈ [M,M + L] ❛♥❞ θ ∈ [x, y]✳

■♥ ♦r❞❡r t♦ ✈❡r✐❢② ✭✐✐✮✱ ✇❡ ✉s❡ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ✇❤✐❝❤ ❣✉❛r❛♥t❡❡s t❤❛t t❤❡ t❡r♠ n log n ✐♥ Sn(f′)(θ) ✐s

✇❡❧❧ ❝♦♥tr♦❧❧❡❞✳ ❚❤❡ ♣r♦❜❧❡♠❛t✐❝ t❡r♠s ❛r❡ U(n, θ) ❛♥❞ V (n, θ)✱ ✇❤✐❝❤ ❞❡♣❡♥❞ ♦♥ t❤❡ ❞✐st❛♥❝❡ ♦❢ T iθ❢r♦♠ t❤❡ s✐♥❣✉❧❛r✐t✐❡s✳ ❲❡ ❞❡✜♥❡ ℓ t♦ ❜❡ s✉❝❤ t❤❛t 1

qℓ+1 log qℓ+1≤ |x − y| < 1

qℓ log qℓ❛♥❞ ❝♦♥s✐❞❡r t✇♦

❝❛s❡s✿ ℓ ∈ KT ♦r ℓ /∈ KT ✭s❡❡ ❉❡✜♥✐t✐♦♥ ✹✳✷✮✳ ❲❤❡♥ ℓ ∈ KT ✱ ✇❡ ✉s❡ ▲❡♠♠❛ ✹✳✻✱ ❜✉t t♦ ❞♦ s♦✱ ✇❡ ♥❡❡❞❣♦♦❞ ❡st✐♠❛t❡s ♦♥ U(qℓ+1, x), V (qℓ+1, x)✳ ❚❤✐s ❝♦♥tr♦❧ ✐s ❣✐✈❡♥ ❜② Pr♦♣♦s✐t✐♦♥ ✺✳✶✱ ✇❤✐❝❤ t❡❧❧s ✉s t❤❛t❡✐t❤❡r ❣♦✐♥❣ ❢♦r✇❛r❞ ♦r ❜❛❝❦✇❛r❞ ✐♥ t✐♠❡✱ ♦♥❡ ❝❛♥ ❝♦♥tr♦❧ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ❤❡♥❝❡ t❤❛t✭✐✮ ✇✐❧❧ ❜❡ s❛t✐s✜❡❞ ❡✐t❤❡r ❢♦r ♣♦s✐t✐✈❡ ♦r ♥❡❣❛t✐✈❡ ✐t❡r❛t❡s✳ ▼♦r❡♦✈❡r✱ ❜② t❤❡ s❛♠❡ r❡❛s♦♥s✱ ✭✹✳✶✻✮ ♦r✭✹✳✶✽✮ ✇✐❧❧ ❜❡ s❛t✐s✜❡❞ ❛♥❞ ❜② ▲❡♠♠❛ ✹✳✻ ❛♥❞ ❤❡♥❝❡ ✇❡ ✇✐❧❧ ❤❛✈❡ ❣♦♦❞ ❝♦♥tr♦❧ ♦❢ ❇✐r❦❤♦✛ s✉♠s ♦❢ t❤❡❞❡r✐✈❛t✐✈❡✱ t❤✉s s❤♦✇✐♥❣ ✭✐✐✮✳ ■♥ t❤❡ s❡❝♦♥❞ ❝❛s❡✱ ♥❛♠❡❧② ✇❤❡♥ ℓ /∈ KT ✱ ❜♦t❤ ✭✐✮ ❛♥❞ ✭✐✐✮ ✐♥ Pr♦♣♦s✐t✐♦♥✹✳✶ ✇✐❧❧ ❤♦❧❞ ❢♦r ♠♦st ♦❢ t❤❡ ♣♦✐♥ts✱ ✐✳❡✳ ♦✉ts✐❞❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤✐❝❤ ❣♦ t♦♦ ❝❧♦s❡ t♦ s♦♠❡ ♦❢ t❤❡s✐♥❣✉❧❛r✐t✐❡s ✭✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ✐♥ ✭✭✺✳✷✶✮ ❛♥❞ ✭✺✳✷✹✮ ❜❡❧♦✇✮✳ ❋♦r r ∈ [qℓ, qℓ+1] ✇❡ ✇❛♥t t❤❡ ♠❛✐♥ t❡r♠✐♥ Sr(f

′) ✭✇❤✐❝❤ ✐s r log r✮ t♦ ❞♦♠✐♥❛t❡ t❤❡ t❡r♠s U(r, x), V (r, x)✳ ◆♦t✐❝❡ t❤❛t ✐❢ r ❣❡ts ❧❛r❣❡r t❤❡ ♠❛✐♥t❡r♠ ✐s ❛❧s♦ ❧❛r❣❡r✱ s♦ t❤❡ ❞❛♥❣❡r ③♦♥❡s ✭✐♥ ✇❤✐❝❤ U(r, x), V (r, x) ❛r❡ t♦♦ ❧❛r❣❡✮ ❛r❡ ❣❡tt✐♥❣ s♠❛❧❧❡r✱ s♦t❤❛t ♦♥❡ ❝❛♥ ❝♦♥tr♦❧ t❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤✐❝❤ ❛r❡ r❡♠♦✈❡❞ ✭s❡❡ ✭✺✳✷✺✮✮✳ ❚❤❡ ❛ss✉♠♣t✐♦♥t❤❛t t❤❡ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ ❢♦r T ✐♠♣❧✐❡s t❤❛t t❤❡ s❡t ♦❢ ℓ /∈ KT ✐s s♠❛❧❧ ❛♥❞ ❤❡♥❝❡ t❤❛t✇❡ ❝❛♥ t❤r♦✇ ❛✇❛② t❤❡ ✉♥✐♦♥ ♦✈❡r ℓ ♦❢ t❤❡ s❡ts ♦❢ ❜❛❞ ♣♦✐♥ts ✭s❡❡ ✭✺✳✷✻✮✮ ❛♥❞ st✐❧❧ ❡♥❞ ✉♣ ✇✐t❤ ❛ s❡t ♦❢❛r❜✐tr❛r❧② ❧❛r❣❡ ♠❡❛s✉r❡✱ ✇❤♦s❡ ♣♦✐♥ts st❛② s✉✣❝✐❡♥t❧② ❢❛r ❢r♦♠ ❛❧❧ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ❤❡♥❝❡ s❛t✐s❢②✭✐✮ ❛♥❞ ✭✐✐✮✳ ❚❤✉s✱ ✇❡ ❝❛♥ ❛♣♣❧② ▲❡♠♠❛ ✹✳✷ ❛♥❞ ❝♦♥❝❧✉❞❡ t❤❡ ♣r♦♦❢✳

✸✻ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✺✳✺✳ ❲❡ r❡♠❛r❦ t❤❛t T ❜② ❞❡✜♥✐t✐♦♥ ♦❢ ❘❛t♥❡r ❉❈ ❛❧s♦ s❛t✐s✜❡s t❤❡ ▼✐①✐♥❣ ❉❈✱✇❤✐❝❤ ✐♠♣❧✐❡s ✐♥ ♣❛rt✐❝✉❧❛r t❤❛t T ✐s ❡r❣♦❞✐❝✳ ❚❤✉s✱ ✐♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❛t (ϕt)t∈R ❤❛s ❙❘✲♣r♦♣❡rt②✐t ✐s ❡♥♦✉❣❤ t♦ ✈❡r✐❢② t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ❲❡ ✇✐❧❧ ❞♦ t❤✐s ❜② ✉s✐♥❣ ▲❡♠♠❛ ✹✳✷ ❛♥❞Pr♦♣♦s✐t✐♦♥ ✺✳✶ ♦♥ ❜❛❝❦✇❛r❞ ♦r ❢♦r✇❛r❞ ❝♦♥tr♦❧ ♦❢ ❞✐st❛♥❝❡ ♦❢ ♦r❜✐ts ❢r♦♠ s✐♥❣✉❧❛r✐t✐❡s✳

❋✐① ε > 0 ✭s♠❛❧❧✮ ❛♥❞ N ∈ N✳ ❚♦ ✈❡r✐❢② Pr♦♣♦s✐t✐♦♥ ✹✳✶✱ ✇❡ ♥❡❡❞ t♦ ❞❡✜♥❡ ❛ κ = κ(ε) t❤❛t ✇❡ s❡t t♦❜❡ κ + ε5✱ ❛ δ = δ(ε,N) ❛♥❞ ❛ s❡t ♦❢ ✏❣♦♦❞✑ ♣♦✐♥ts X ′ ✇✐t❤ λ(X ′) > 1− ε✳ ❙✐♥❝❡ T s❛t✐s✜❡s t❤❡ ❘❛t♥❡r❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✇✐t❤ τ, ξ, η s❛t✐s❢②✐♥❣ ✭✺✳✶✽✮✱ ❜② t❤❡ ❉❡✜♥✐t✐♦♥ ✹✳✷ ♦❢ ❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥❛♥❞ ❜② ❈♦r♦❧❧❛r② ✹✳✽✱ ✇❡ ❤❛✈❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ s❡r✐❡s ❛r❡ s✉♠♠❛❜❧❡✿

✭✺✳✶✾✮∑

ℓ/∈KT

σηℓ <∞,∑

ℓ/∈KT

λ(Σℓ(T )) <∞,

✇❤❡r❡ KT = {ℓ ∈ N : qℓ+L ≤ qℓ

σξ′

ℓ

} ✭✇❤❡r❡ ξτ ′ +10−3 > ξ′ > ξ

τ ′ ✐s ❛ s♠❛❧❧✮✳ ❍❡♥❝❡✱ t❤❡r❡ ❡①✐sts l0(ε) s✉❝❤

t❤❛t ✐❢ ✇❡ s❡t

✭✺✳✷✵✮ Z1 :=⋃

l /∈KT ,ℓ≥ℓ0

2Σℓ(T ), t❤❡♥ λ(Z1) <ε

3, ❛♥❞

∑

l /∈KT , ℓ≥ℓ0

σηℓ <ε

6ν2|A|.

❋✐① l /∈ KT , l ≥ l0 ❛♥❞ k ∈ {0, . . . ,[ql+1

qℓ

]}✳ ❉❡✜♥❡ t❤❡ s❡t

✭✺✳✷✶✮ Jkℓ +⋃

α∈A

(k+1)qℓ−1⋃

i=kqℓ

T−i[−1

(k + 1)qℓ(log(k + 1)qℓ)ξ+ lα, lα +

1

(k + 1)qℓ(log(k + 1)qℓ)ξ].

◆♦t✐❝❡ t❤❛t

✭✺✳✷✷✮ λ(Jkℓ ) ≤2|A|

(k + 1)(log qℓ)ξ.

▼♦r❡♦✈❡r ❜② ✭✹✳✶✷✮ ❢♦r ℓ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ✇❡ ❤❛✈❡

✭✺✳✷✸✮6|A|

(log qℓ)ξlog ‖Aℓ‖ ≤ σηl .

◆♦✇ ❞❡✜♥❡

✭✺✳✷✹✮ Jℓ +

[qℓ+1qℓ

]+1⋃

k=0

Jkℓ .

❲❡ ❤❛✈❡ ❜② ✭✺✳✷✷✮

✭✺✳✷✺✮ λ(Jℓ) ≤2|A|

(log qℓ)ξ

[qℓ+1qℓ

]+1∑

k=0

1

k≤

2|A|

(log qℓ)ξ2 log(

[qℓ+1

qℓ

]+ 1) ≤

6|A|

(log qℓ)ξlog ‖Aℓ‖

✭✺✳✷✸✮

≤ σηℓ

❲❡ ❞❡✜♥❡

✭✺✳✷✻✮ Z2 +⋃

l /∈KT ,l≥l0

Jℓ.

◆♦t✐❝❡ t❤❛t ❜② t❤❡ ❛❜♦✈❡ ❝♦♠♣✉t❛t✐♦♥s ❛♥❞ ✭✺✳✷✵✮ ✇❡ ❤❛✈❡ λ(Z2) ≤ε3 ✳ ❋✐♥❛❧❧②✱ ✇❡ ❞❡✜♥❡

X ′+ Zc1 ∩ Z

c2 ∩ (

ε

8, 1−

ε

8).

◆♦t✐❝❡ t❤❛t s✐♥❝❡ λ(Z1), λ(Z2) < ε/3✱ ✇❡ ❤❛✈❡ λ(X ′) > 1 − ε✳ ❲❡ ✇✐❧❧ ♣r♦✈❡ ❘❛t♥❡r ✭s❡❡ Pr♦♣♦s✐t✐♦♥✹✳✶✮ ❢♦r ♣❛✐rs ♦❢ ♣♦✐♥ts ❢r♦♠ X ′✳ ▲❡t ♥♦✇

✭✺✳✷✼✮ ℓa = max(N2 + 1

ε4, 1/ε, l0, ℓ1 + 1, ℓ′),

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✸✼

✇❤❡r❡ ℓ1(ε2) ✐s s✉❝❤ t❤❛t t❤❡ ❡st✐♠❛t❡s ✐♥ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ❛♥❞ ▲❡♠♠❛ ✹✳✻ ❤♦❧❞ ❢♦r ℓ ≥ ℓ1 ❛♥❞ ℓ

′ ❝♦♠❡s❢r♦♠ Pr♦♣♦s✐t✐♦♥ ✺✳✶✳ ❉❡✜♥❡

✭✺✳✷✽✮ δ + min(1

ℓ2a, ε2).

❲❡ ✇✐❧❧ s❤♦✇ t❤❛t ❛♥② x, y ∈ X ′ ✇✐t❤ |x− y| < δ s❛t✐s❢② ✭✐✮ ♦r ✭✐✐✮ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳▲❡t r ∈ N ❜❡ t❤❡ ✉♥✐q✉❡ ♥✉♠❜❡r s✉❝❤ t❤❛t

✭✺✳✷✾✮1

(C− − C+)(r + 1) log(r + 1)< y − x ≤

1

(C− − C+)r log r.

▲❡t ♥♦✇ ℓ ∈ N ❜❡ t❤❡ ✉♥✐q✉❡ ♥✉♠❜❡r s✉❝❤ t❤❛t qℓ ≤ r < qℓ+1 ✭♥♦t❡ t❤❛t ❜② ✭✺✳✷✽✮ ℓa < ℓ✮✳❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝❛s❡s✿

❈❛s❡ ✶✳ ℓ ∈ KT ✭✐♥ ♣❛rt✐❝✉❧❛r qℓ+L ≤ c10qℓ(log qℓ)

ξ ✇❤❡r❡ c ✐s ❝♦♠♠✐♥❣ ❢r♦♠ Pr♦♣♦s✐t✐♦♥ ✺✳✶✮✳■♥ t❤✐s ❝❛s❡✱ s✐♥❝❡ x ∈ X ′✱ ✇❡ ❝❛♥ ✉s❡ Pr♦♣♦s✐t✐♦♥ ✺✳✶ ✭✇✐t❤ ℓ+ 1✮ t♦ ❣❡t ✭❲▲❖● ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛tL ≥ 1✮

✭✺✳✸✵✮ d({lα, rα : α ∈ A}, {T ix : 0 ≤ i < qℓ+1}) >c

qℓ+L≥

2

qℓ(log qℓ)ξ,

♦r

✭✺✳✸✶✮ d({lα, rα : α ∈ A}, {T ix : −qℓ+1 ≤ i < 0}) >c

qℓ+L≥

2

qℓ(log qℓ)ξ.

■❢ ✭✺✳✸✵✮ ❤♦❧❞s✱ ✇❡ s❤♦✇ ✭✐✮✱ ✐❢ ✭✺✳✸✶✮ ❤♦❧❞s ✇❡ s❤♦✇ ✭✐✐✮✳ ❙✐♥❝❡ t❤❡ ♣r♦♦❢s ✐♥ ❜♦t❤ ❝❛s❡s ❛r❡ ❛♥❛❧♦❣♦✉s✱✇❡ ✇✐❧❧ ❝♦♥❞✉❝t t❤❡ ♣r♦♦❢ ❛ss✉♠✐♥❣ ✭✺✳✸✵✮ ❤♦❧❞s✳

▲❡t

✭✺✳✸✷✮ M + min(r, (1− ε4)qℓ+1) ❛♥❞ L = [ε5M ] + 1,

✭s♦ t❤❛t L/M ≥ κ ❛♥❞ M + L < qℓ+1✮✳ ◆♦t✐❝❡ t❤❛t ‖x− y‖ < δ✭✺✳✷✽✮< ε✳ ▼♦r❡♦✈❡r

M ≥ L > ε4M ≥ ε4qℓ > ε4ℓ > ε4ℓa > N,

✭t❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ❜② ✭✺✳✷✼✮✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ▲❡♠♠❛ ✹✳✷ ❛r❡ s❛t✐s✜❡❞ ❢♦r x, y,M,L✳❍❡♥❝❡✱ t♦ s❤♦✇ ✭✐✮ ✐♥ Pr♦♣♦s✐t✐♦♥ ✹✳✶✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ✈❡r✐❢② t❤❛t ✭✹✳✶✮✱✭✹✳✷✮✱✭✹✳✸✮ ✐♥ ▲❡♠♠❛ ✹✳✷ ❛r❡s❛t✐s✜❡❞✳

❚♦ s❤♦✇ ✭✹✳✶✮ ♥♦t✐❝❡ ✜rst t❤❛t ❜② ✭✺✳✷✾✮ ❛♥❞ r ≥ qℓ✱ |y − x| < 1(C−−C+)qℓ log qℓ

✳ ❲❡ ❤❛✈❡ ❢♦r ❡✈❡r②

θ ∈ [x, y]

✭✺✳✸✸✮ d({lα : α ∈ A}, {T iθ : 0 ≤ i < qℓ+1}

)≥

d({lα : α ∈ A}, {T ix : 0 ≤ i < qℓ+1}

)− |θ − x|

✭✺✳✸✵✮>

2

qℓ(log qℓ)ξ−

1

(C− − C+)qℓ log qℓ>

1

qℓ(log qℓ)ξ

t❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ❢♦❧❧♦✇s s✐♥❝❡ ξ < 1 ❛♥❞ ℓ ✐s ❧❛r❣❡✳ ❚❤✐s ❛♥❞ M + L < qℓ+1 ❣✐✈❡s ✭✹✳✶✮✳◆♦t✐❝❡ t❤❛t ❜② ✭✺✳✸✸✮✱ ❢♦r ❡✈❡r② θ ∈ [x, y] ✭✹✳✶✻✮ ❤♦❧❞s✳ ❙♦ ✉s✐♥❣ ✭✹✳✶✼✮ ✭[M,M + L] ⊂ [qℓ, qℓ+1)✮✱ ✇❡

❣❡t t❤❛t ❢♦r ❡✈❡r② (r, θ) ∈ [M,M + L]× [x, y]

✭✺✳✸✹✮ 0 < (C− − C+ − ε2)r log r ≤ Sr(f′)(θ) ≤ (C− − C+ + ε2)r log r,

s♦ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ❛✉t♦♠❛t✐❝❛❧❧② ❣✐✈❡s ✭✹✳✷✮✳◆♦✇ ✇❡ s❤♦✇ ✭✹✳✸✮✳ ❇② ✭✺✳✸✹✮ ✇❡ ❣❡t ❢♦r ❡✈❡r② (r, θ) ∈ [M,M + L]× [x, y]

✭✺✳✸✺✮ (C− − C+ − ε2)M logM ≤ Sr(f′)(θ) ≤ (C− − C+ + ε2)(M + L) log(M + L).

❇✉t ❜② ✭✺✳✸✷✮ ❛♥❞ ✭✺✳✷✾✮ ✇❡ ❣❡t

M ≥ (1− ε3)(r + 1) ❛♥❞ M + L ≤ (1 + ε4)r.

✸✽ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

P❧✉❣✐♥❣ t❤✐s ✐♥t♦ ✭✺✳✸✺✮ ❛♥❞ ✉s✐♥❣ ✭✺✳✷✾✮ ❣✐✈❡s ✭✹✳✸✮✳ ❙♦ ❜② ▲❡♠♠❛ ✹✳✷ ✭✐✮ ✐s s❛t✐s✜❡❞✳ ❚❤✐s ✜♥✐s❤❡s t❤❡♣r♦♦❢ ✐♥ ❈❛s❡ ✶✳

❈❛s❡ ✷✳ ℓ /∈ KT ✳◆♦t✐❝❡ ✜rst t❤❛t ❢♦r ❡✈❡r② j ∈ {0, ..., [σℓqℓ + 1]}✱ ✇❡ ❤❛✈❡

✭✺✳✸✻✮ lα /∈ [T jx, T jy] ❢♦r ❡✈❡r② α ∈ A.

■♥❞❡❡❞✱ ♥♦t✐❝❡ t❤❛t σℓ log qℓ → 0 ❛s ℓ→ +∞✳ ❚❤✐s✱ nℓ ❜❡✐♥❣ ❛ ❜❛❧❛♥❝❡❞ t✐♠❡ ❛♥❞ ✭✺✳✷✾✮ ❣✐✈❡

✭✺✳✸✼✮ σℓI(nℓ) ≥

νσℓqℓ

≥100

qℓ log qℓ> 2|y − x|.

❇✉t x ∈ X ′ ⊂ Zc1 ⊂ (2Σ+ℓ (T ))

c✱ s♦ ❢♦r ❡✈❡r② j ∈ {0, ..., [σℓqℓ + 1]} ❛♥❞ ❡✈❡r② α ∈ A✱

✭✺✳✸✽✮ d(T jx, lα) ≥ 2σℓI(nℓ) > |y − x|,

✇❤✐❝❤ ❣✐✈❡s ✭✺✳✸✻✮✳ ❲❡ ❝❧❛✐♠ t❤❛t ❢♦r ❡✈❡r② θ ∈ [x, y]

✭✺✳✸✾✮ θ /∈ Σ+ℓ (T ).

▲❡t ✉s ♣r♦✈❡ ✭✺✳✸✾✮ ❜② ❝♦♥tr❛❞✐❝t✐♦♥✿ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ Σ+ℓ (T )✱ ✐❢ ✭✺✳✸✾✮ ❢❛✐❧s✱ ✐t ✇♦✉❧❞ ♠❡❛♥ t❤❛t

t❤❡r❡ ❡①✐st i ∈ {0, ..., [σℓqℓ + 1]} ❛♥❞ α ∈ A✱ s✉❝❤ t❤❛t d(T iθ, lα) ≤ σℓI(nℓ) ✭❝❤♦♦s❡ i t♦ ❜❡ t❤❡ s♠❛❧❧❡st

♦♥❡ ✇✐t❤ t❤✐s ♣r♦♣❡rt②✮✳ ❙♦

d(T ix, lα) ≤ d(T iθ, lα) + d(T ix, T iθ) ≤ σℓI(nℓ) + |y − x|

✭✺✳✸✼✮< 2σℓI

(nℓ),

❛ ❝♦♥tr❛❞✐❝t✐♦♥ ✇✐t❤ ✭✺✳✸✽✮✳ ❙♦ ✭✺✳✸✾✮ ❤♦❧❞s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✱ ❢♦r ❡✈❡r② θ ∈ [x, y]

✭✺✳✹✵✮ U(m, θ), V (m, θ) ≤ 2m(logm)ξ,

❢♦r ❡✈❡r② m ∈ [qℓ, (1 + ε)r]✳ ▲❡t ✉s s❤♦✇ t❤❛t ❢♦r ❡✈❡r② m ∈ [qℓ, (1 + ε)r]✱ U(m, θ) > 2m(logm)ξ ✭t❤❡♣r♦♦❢ ♦❢ V (m, θ) > 2m(logm)ξ ✐s ❛♥❛❧♦❣♦✉s✮✳

❚❤✐s ❢♦❧❧♦✇s ❜② t❤❡ ❢❛❝t t❤❛t x ∈ X ′ ⊂ Zc2✭✺✳✷✻✮⊂ Jℓ✳ ■♥❞❡❡❞✱ ❧❡t 1 ≤ k ∈≤

[qℓ+1

qℓ

]+ 1 ❜❡ s✉❝❤ t❤❛t

kqℓ ≤ m < (k + 1)qℓ✳ ❚❤❡♥✱ ❜② ✭✺✳✷✹✮✱ ✭✺✳✷✶✮✱ t❤❡ ❢❛❝t t❤❛t x ∈ Jkℓ ✱ ✇❡ ❣❡t

minα∈A

d({x, ..., Tmx}, lα) ≥1

(k + 1)qℓ(log(k + 1)qℓ)ξ≥

1

2m(log 2m)ξ.

❚❤❡r❡❢♦r❡✱ ❛♥❞ s✐♥❝❡ m ≤ (1 + ε)r ❛♥❞ ξ < 1✱ ✇❡ ❤❛✈❡

minα∈A

d({θ, ..., Tmθ}, lα)✭✺✳✷✾✮

≥ minα∈A

d({x, ..., Tmx}, lα)−1

(C− − C+)r log r≥

1

2m(logm)ξ,

s♦ ✭✺✳✹✵✮ ❤♦❧❞s✳◆♦✇ ❞❡✜♥❡ M + max(r, (1− ε4)qℓ+1)✱ L + [ε5M ] + 1✳ ❋r♦♠ t❤✐s ♣♦✐♥t t❤❡ ♣r♦♦❢ ✐s ❛♥❛❧♦❣♦✉s t♦ t❤❡

♣r♦♦❢ ♦❢ ❈❛s❡ ✶✳ ✿ ❲❡ ✈❡r✐❢② ❛ss✉♠♣t✐♦♥s ✭✹✳✶✮✱ ✭✹✳✷✮ ❛♥❞ ✭✹✳✸✮ ✐♥ ▲❡♠♠❛ ✹✳✷✳ ❇② ✭✺✳✹✵✮ ✇❡ ❣❡t t❤❛t✭✹✳✶✮ ❤♦❧❞s✳

▼♦r❡♦✈❡r✱ ❜② ✭✺✳✸✾✮ ❛♥❞ ✭✺✳✹✵✮ ❛♥❞ Pr♦♣♦s✐t✐♦♥ ✹✳✹ ✐t ❢♦❧❧♦✇s t❤❛t ❢♦r ❡✈❡r② r, θ ∈ [M,M +L]× [x, y]

(C− − C+ − ε2)r log r ≤ Sr(f′)(θ) ≤ (C− − C+ + ε2)r log r.

◆♦✇ s✐♥❝❡ M,L ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ ❈❛s❡ ✶✳ ❛♥❞ t❤❡ ❛❜♦✈❡ ❡st✐♠❛t❡ ✐s t❤❡ s❛♠❡ ❛s ✭✺✳✸✹✮ ✇❡ ✈❡r✐❢②✭✹✳✷✮ ❛♥❞ ✭✹✳✸✮ r❡♣❡❛t✐♥❣ t❤❡ r❡st ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ❈❛s❡ ✶✳ ❚❤✐s ✜♥✐s❤❡s t❤❡ ♣r♦♦❢ ✐♥ ❈❛s❡ ✷✳

�

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✸✾

✺✳✸✳ ❈♦♥❝❧✉s✐♦♥s✳ ■♥ t❤✐s ✜♥❛❧ s❡❝t✐♦♥ ✇❡ ❝♦♥❝❧✉❞❡ ❜② ❣✐✈✐♥❣ t❤❡ ♣r♦♦❢s ♦❢ ❛❧❧ t❤❡ r❡s✉❧ts st❛t❡❞ ✐♥ t❤❡✐♥tr♦❞✉❝t✐♦♥✳ ❚❤❡ r❡s✉❧ts ♣r♦✈❡❞ s♦ ❢❛r ✐♠♠❡❞✐❛t❡❧② ❣✐✈❡ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✹✱ ♥❛♠❡❧② s❤♦✇ t❤❛ts♣❡❝✐❛❧ ✢♦✇s ✉♥❞❡r ❢✉♥❝t✐♦♥s ✇✐t❤ ❧♦❣❛r✐t❤♠✐❝ ❛s②♠♠❡tr✐❝ s✐♥❣✉❧❛r✐t✐❡s ❤❛✈❡ t❤❡ ❙❘✲♣r♦♣❡rt② ❢♦r ❛✳❡✳■❊❚✿

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✹✳ ▲❡t τ ∈ (1, 16/15)✱ τ ′ ∈ (15/16, 1)✱ ξ′ > 99/100✱ η′ > 3/4✳ ❋♦r ❡❛❝❤ ✐rr❡❞✉❝✐❜❧❡❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t✉♠ π ❛♥❞ ❢♦r ▲❡❜❡s❣✉❡ ❛✳❡✳ λ ∈ ∆d✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ■❊❚ T = (λ, π) s❛t✐s✜❡s t❤❡❙✉♠♠❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✇✐t❤ ❡①♣♦♥❡♥ts (τ ′, ξ′, η′) ❜② ❈♦r♦❧❧❛r② ✹✳✾✳ ❍❡♥❝❡✱ ❜② Pr♦♣♦s✐t✐♦♥ ✺✳✺✱ t❤❡s♣❡❝✐❛❧ ✢♦✇ (ϕt)t∈R ♦✈❡r T ❛♥❞ ✉♥❞❡r f ∈ AsymLogSing(T ) ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✳ �

▲❡t ✉s ♥♦✇ ♣r♦✈❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡s✉❧t ♦♥ t❤❡ s✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❧♦❝❛❧❧②❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s✱ ♥❛♠❡❧② ❈♦r♦❧❧❛r② ✶✳✻✳

Pr♦♦❢ ♦❢ ❈♦r♦❧❧❛r② ✶✳✻✳ ❋♦r ❛♥② ✜①❡❞ ❣❡♥✉s g ≥ 1✱ ❝♦♥s✐❞❡r t❤❡ ♦♣❡♥ s❡t U¬min ♦❢ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥✢♦✇s ♦♥ ❛ s✉r❢❛❝❡ S ♦❢ ❣❡♥✉s g ✇✐t❤ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts ✇❤✐❝❤ ❤❛✈❡ ❛ s❛❞❞❧❡ ❧♦♦♣ ❤♦♠♦❧♦❣♦✉st♦ ③❡r♦✳ ❊q✉✐✈❛❧❡♥t❧②✱ t❤❡s❡ ❛r❡ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✇✐t❤ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts ✇❤✐❝❤ ❤❛✈❡❛t ❧❡❛st ♦♥❡ ♣❡r✐♦❞✐❝ ❝♦♠♣♦♥❡♥t✳ ❆s ❡①♣❧❛✐♥❡❞ ❜② ❘❛✈♦tt✐ ✐♥ ❬✸✺❪ ✭s❡❡ ❙❡❝t✐♦♥ ✷ ❛♥❞ ❙❡❝t✐♦♥ ✸✮✱ t❤❡r❡❡①✐sts ❛♥ ♦♣❡♥ ❛♥❞ ❞❡♥s❡ s❡t U′

¬min ⊂ U¬min ✭t❤✐s ✐s ❞❡♥♦t❡❞ ❜② A′s,l ✐♥ ❬✸✺❪✱ s❡❡ ◆♦t❛t✐♦♥ ✸✳✸✮ s✉❝❤ t❤❛t

❛♥② ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ❛ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ (ϕt)t∈R ✐♥ U′¬min ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ s♣❡❝✐❛❧

✢♦✇ ♦✈❡r ❛♥ ■❊❚ T = (λ, π) ✉♥❞❡r ❛ r♦♦❢ f ∈ AsymLogSing(T ) ✇❤❡r❡ π ✐s ✐rr❡❞✉❝✐❜❧❡✳ ❋✉rt❤❡r♠♦r❡✱ ❜②❘❡♠❛r❦ ✷✳✶✱ ❛ ♣r♦♣❡rt② ✇❤✐❝❤ ❤♦❧❞s ❢♦r ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t ♦❢ ■❊❚s ♦♥ ❛♥② ♥✉♠❜❡r ♦❢ ✐♥t❡r✈❛❧s ❛❧s♦ ❤♦❧❞s❢♦r t❤❡ s♣❡❝✐❛❧ ✢♦✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❡❛❝❤ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ❛ ❢✉❧❧ ♠❡❛s✉r❡ ♦❢ ✢♦✇s ✐♥ U′

¬min✳ ❚❤✉s✱❜② ❚❤❡♦r❡♠ ✶✳✹✱ ❡❛❝❤ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ❛ t②♣✐❝❛❧ ✢♦✇ ✐♥ U′

¬min ❛❞♠✐ts ❛ r❡♣r❡s❡♥t❛t✐♦♥ ❛s ❛ s♣❡❝✐❛❧✢♦✇ ✇❤✐❝❤ ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✳ ❙✐♥❝❡ t❤❡ s♣❡❝✐❛❧ ✢♦✇ r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ♠❡tr✐❝❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ t❤❡r❡str✐❝t✐♦♥s ♦❢ (ϕt)t∈R t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ❛♥❞ t❤❡ ❙❘✲♣r♦♣❡rt② ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✐♥✈❛r✐❛♥t ✭s❡❡ ▲❡♠♠❛ ❆✳✶ ✐♥ ❆♣♣❡♥❞✐① ❆✳✶✮✱ ✐t ❢♦❧❧♦✇s ❢♦r ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t ♦❢ ✢♦✇s ✐♥ U′

¬min✱ ❡❛❝❤r❡str✐❝t✐♦♥ ♦❢ t❤❡ ✢♦✇ t♦ ❛ ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ❤❛s t❤❡ ❙❘✲❘❛t♥❡r ♣r♦♣❡rt②✳ �

❋r♦♠ t❤❡ ❙❘✲♣r♦♣❡rt②✱ ✇❡ ❛♥ ♥♦✇ ❞❡❞✉❝❡ t❤❡ r❡s✉❧ts ♦♥ ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs ✐♥ t❤❡ s❡t ✉♣ ♦❢ s♣❡❝✐❛❧✢♦✇s ✭❚❤❡♦r❡♠ ✶✳✷✮ ❛♥❞ ✜♥❛❧❧② ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✭❚❤❡♦r❡♠ ✶✳✶✮✳

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✷✳ ❋✐① ❛♥② ✐rr❡❞✉❝✐❜❧❡ ♣❡r♠✉t❛t✐♦♥ π✳ ❈♦♥s✐❞❡r t❤❡ s♣❡❝✐❛❧ ✢♦✇ (ϕt)t∈R ♦✈❡r T =(λ, π) ❛♥❞ ✉♥❞❡r ❛ r♦♦❢ f ∈ AsymLogSing(T )✳ ❇② ❚❤❡♦r❡♠ ✶✳✹✱ ❢♦r ▲❡❜❡s❣✉❡ ❛✳❡✳ λ ∈ ∆d✱ (ϕt)t∈R ❤❛st❤❡ ❙❘✲♣r♦♣❡rt② ❛♥❞ ❤❡♥❝❡ ✐♥ ♣❛rt✐❝✉❧❛r ❛❧s♦ t❤❡ ❙❲❘✲♣r♦♣❡rt② ✭✇❤✐❝❤ ✐s ✇❡❛❦❡r✱ r❡❝❛❧❧ t❤❡ ❉❡✜♥✐t✐♦♥s✷✳✷ ❛♥❞ ✷✳✸✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❜② Pr♦♣♦s✐t✐♦♥ ✸✳✸ ♣r♦✈❡❞ ✐♥ ❬✹✶❪ ❛♥❞ ❚❤❡♦r❡♠ ✸✳✷ ✭s❡❡ ❬✹✶✱ ✸✺❪✮✱ ❢♦r▲❡❜❡s❣✉❡ ❛✳❡✳ λ ∈ ∆d ✇❡ ❛❧s♦ ❤❛✈❡ t❤❛t (ϕt)t∈R ✐s ♠✐①✐♥❣✳ ❚❤✉s✱ ❢♦r ❛ ❢✉❧❧ ♠❡❛s✉r❡ s❡t ♦❢ λ ∈ ∆d✱(ϕt)t∈R ✐s ♠✐①✐♥❣ ❛♥❞ ❤❛s t❤❡ ❙❲❘✲♣r♦♣❡rt②✱ ✇❤✐❝❤✱ ❜② ❚❤❡♦r❡♠ ✷✳✹✱ ✐♠♣❧✐❡s t❤❛t (ϕt)t∈R ✐s ❛❧s♦ ♠✐①✐♥❣♦❢ ❛❧❧ ♦r❞❡rs✳ �

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✶✳ ❇② ❈♦r♦❧❧❛r② ✶✳✻✱ ❢♦r ❛♥② g ≥ 1 t❤❡r❡ ❡①✐sts ❛♥ ♦♣❡♥ ❛♥❞ ❞❡♥s❡ s❡t U′¬min ✐♥ t❤❡

♦♣❡♥ s❡t U¬min ♦❢ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇s ♦♥ ❛ s✉r❢❛❝❡ S ♦❢ ❣❡♥✉s g ≥ 1 ✭✇❤✐❝❤ ✇❡ r❡❝❛❧❧ ❝♦♥s✐sts♦❢ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✇✐t❤ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts ✇❤✐❝❤ ❤❛✈❡ ❛ s❛❞❞❧❡ ❧♦♦♣ ❤♦♠♦❧♦❣♦✉s t♦ ③❡r♦✮s✉❝❤ t❤❛t ❛♥② ♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ❛ t②♣✐❝❛❧ ❧♦❝❛❧❧② ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ (ϕt)t∈R ✐♥ U′

¬min ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✳ ❋✉rt❤❡r♠♦r❡✱ ❜② Pr♦♣♦s✐t✐♦♥ ✸✳✸✱ ❚❤❡♦r❡♠ ✸✳✷ ❛♥❞ ❘❡♠❛r❦ ✷✳✶✱ ♦♥❡ ❝❛♥ ❛❧s♦ ❛ss✉♠❡ ❜② t❤❡s❛♠❡ ❛r❣✉♠❡♥ts ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❈♦r♦❧❧❛r② ✶✳✻ t❤❛t ❢♦r t②♣✐❝❛❧ (ϕt)t∈R ✐♥ U′

¬min t❤❡ r❡str✐❝t✐♦♥ t♦ ❡❛❝❤♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t ✐s ❛❧s♦ ♠✐①✐♥❣✳ ❚❤✉s✱ ❜② ❚❤❡♦r❡♠ ✷✳✹✱ (ϕt)t∈R ✐s ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs ♦♥ ❡❛❝❤♠✐♥✐♠❛❧ ❝♦♠♣♦♥❡♥t✳ �

❲❡ ❝♦♥❝❧✉❞❡ ❜② ♣r♦✈✐♥❣ ❈♦r♦❧❧❛r② ✶✳✻ ✇❤✐❝❤ ✐s ❛ str❡❣❤t❤❡♥✐♥❣ ♦❢ t❤❡ ♠❛✐♥ r❡s✉❧t ❜② ❋❛②❛❞ ❛♥❞t❤❡ ✜rst ❛✉t❤♦r ❬✶✵❪✳ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ t❤❡ ♦❜s❡r✈❛t✐♦♥ t❤❛t ✇❡ ❝❛♥ ❛❞❞ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦❢ t❤❡❢✉♥❝t✐♦♥ ❛s ♠❛r❦❡❞ ♣♦✐♥ts✱ s♦ t❤❛t ✇❡ ❣❡t ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✐♥❣✉❧❛r✐t✐❡s ❛t t❤❡❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ ❛ ■❊❚ ✇✐t❤ ❢❛❦❡ s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ♦♥ ❛ ❋✉❜✐♥✐ ❛r❣✉♠❡♥t✳

Pr♦♦❢ ♦❢ ❈♦r♦❧❧❛r② ✶✳✻✳ ❆ss✉♠❡ ❜② ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t t❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s ❢❛❧s❡❀ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ s❡t ♦❢♣♦s✐t✐✈❡ ♠❡❛s✉r❡ A ⊂ [0, 1] ❛♥❞ ❛ s❡t ♦❢ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ X ⊂ [0, 1]d s✉❝❤ t❤❛t t❤❡ s♣❡❝✐❛❧ ✢♦✇ ♦✈❡r Rα✇✐t❤ α ∈ A ✉♥❞❡r ❛ r♦♦❢ f ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❉❡✜♥✐t✐♦♥ ✷✳✶✮ ❛t x0 + 0 < x1 < · · · <xd < xd+1 + 1 ❣✐✈❡♥ ❜② x + (x1, . . . , xd) ∈ X ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ❙❘✲♣r♦♣❡rt②✳ ❲❡ ❝❛♥ ❝❤♦♦s❡ (α0✱

✹✵ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

x0) ∈ A × X t♦ ❜❡ ❛ ▲❡❜❡s❣✉❡ ❞❡♥s✐t② ♣♦✐♥t s✉❝❤ t❤❛t α 6= xi ❢♦r ❛♥② 0 ≤ i ≤ d + 1✱ s✐♥❝❡ ❜♦t❤ ❛r❡❢✉❧❧ ♠❡❛s✉r❡ ❝♦♥❞✐t✐♦♥s✳ ❙❛② t❤❛t x0i−1 < 1− α0 < x0i ❢♦r 1 ≤ i ≤ d+ 1✳ ❚❤✐s r❡❧❛t✐♦♥ ✇✐❧❧ ❛❧s♦ ❜❡ tr✉❡

❢♦r (α, x) s✉✣❝✐❡♥t❧② ❝❧♦s❡ t♦ (α0, x0)✳ ❋♦r ❛❧❧ t❤❡s❡ ♣❛r❛♠❡t❡rs (α, x)✱ ❜② t❤✐♥❦✐♥❣ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡sxi ❛s ❛s ♠❛❦❡❞ ♣♦✐♥ts✱ ✇❡ ❝❛♥ t❤✐♥❦ ♦❢ t❤❡ r♦t❛t✐♦♥s Rα ❛s ■❊❚s ♦♥ d+ 1 ✐♥t❡r✈❛❧s✱ ✇❤♦s❡ ❧❡♥❣t❤s ❛♥❞❝♦♠❜✐♥❛t♦r✐❛❧ ❞❛t❛ ❛r❡ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ❜②

✭✺✳✹✶✮ λ + (x1, x2 − x1, . . . , xi−1 − xi−2, 1− α− xi−1, xi − (1− α), xi+1 − xi, . . . , 1− xd) ,

πrot,it = (1, 2, . . . , i, i+ 1, . . . , d+ 1),

πrot,ib = (i, . . . , d+ 1, 1, 2, . . . , i− 1).

❲❡ r❡♠❛r❦ t❤❛t πrot,i = (πrot,it , πrot,ib ) ✐s ✐rr❡❞✉❝✐❜❧❡✳ ❇② ▲❡❜❡s❣✉❡ ❞❡♥s✐t② ❚❤❡♦r❡♠✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛♥ s❡t

E ⊂ A×X ⊂ [0, 1]d+1 ♦❢ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ s✉❝❤ t❤❛t ✭✺✳✹✶✮ ❢♦r ❛❧❧ (α, x) ✐♥ E✳ ❙✐♥❝❡ t❤❡ ♠❛♣ (α, x) → λ❣✐✈❡♥ ❜② ✭✺✳✹✶✮ ✐s ❧✐♥❡❛r✱ t❤✐s ❣✐✈❡s ❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ s❡t ♦❢ λ ∈ [0, 1]d+1 s✉❝❤ t❤❛t t❤❡ s♣❡❝✐❛❧ ✢♦✇ ♦✈❡rT = (λ, πrot,i) ✇✐t❤ f ∈ AsymLog(T ) ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ❙❘✲♣r♦♣❡rt②✱ ❤❡♥❝❡ ❝♦♥tr❛❞✐❝t✐♥❣ ❚❤❡♦r❡♠✶✳✹✳ ❚❤✐s ❝♦♥❝❧✉❞❡s t❤❡ ♣r♦♦❢✳ �

❆♣♣❡♥❞✐① ❆✳

■♥ t❤✐s ❆♣♣❡♥❞✐① ✇❡ ✐♥❝❧✉❞❡✱ ❢♦r ❝♦♥✈❡♥✐❡♥❝❡ ♦❢ t❤❡ r❡❛❞❡r✱ t❤❡ ♣r♦♦❢s ♦❢ t✇♦ r❡s✉❧ts ✉s❡❞ ✐♥ t❤❡♣r❡✈✐♦✉s s❡❝t✐♦♥s✱ ♥❛♠❡❧② t❤❡ ♣r♦♦❢ t❤❛t t❤❡ ❙✇✐t❝❤❛❜❧❡ ❘❛t♥❡r ♣r♦♣❡rt② ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t✭✐♥ ❙❡❝t✐♦♥ ❆✳✶✮ ❛♥❞ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛ ❢r♦♠ ❬✶✹❪ ✇❤✐❝❤ ❛❧❧♦✇s t♦ ❝♦♥tr♦❧ ❞✐st❛♥❝❡s ❛♠♦♥❣ ❞✐s❝♦♥✲t✐♥✉✐t✐❡s ♦❢ ❛♥ ■❊❚ ✐♥ t❡r♠s ♦❢ t❤❡ ❧❡♥❣❤t ♦❢ t❤❡ ✐♥❞✉❝✐♥❣ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ♥❡①t ❜❛❧❛♥❝❡❞ ❘❛✉③②✲❱❡❡❝❤t✐♠❡ ✭✐♥ ❙❡❝t✐♦♥ ❆✳✷✮✳

❆✳✶✳ ❘❛t♥❡r ♣r♦♣❡rt✐❡s ❛r❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t✳ ■♥ t❤✐s ❆♣♣❡♥❞✐① ✇❡ ✐♥❝❧✉❞❡ ❢♦r ❝♦♠✲♣❧❡t❡♥❡ss t❤❡ ♣r♦♦❢ t❤❛t t❤❡ ❙❘✲♣r♦♣❡rt② ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥✈❛r✐❛♥t ✭t❤❡ s❛♠❡ ❤♦❧❞s ❢♦r ♦t❤❡r ❘❛t♥❡r♣r♦♣❡rt✐❡s ✇✐t❤ t❤❡ s❡t P ❜❡✐♥❣ ✜♥✐t❡✮✳

▲❡♠♠❛ ❆✳✶✳ ▲❡t (X, (Tt),A , µ, dT ) ❛♥❞ (Y, (St),B, ν, dS) t✇♦ ♠❡❛s✉r❛❜❧② ✐s♦♠♦r♣❤✐❝ ♠❡❛s✉r❡ ♣r❡✲s❡r✈✐♥❣ ✢♦✇s✳ ❚❤❡♥✱ ✐❢ (Tt) ❤❛s t❤❡ ❙❘✲♣r♦♣❡rt②✱ ❛❧s♦ (St) ❞♦❡s✳

Pr♦♦❢✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② ψ : X → Y t❤❡ ♠❡❛s✉r❛❜❧❡ ✐s♦♠♦r♣❤✐s♠✳ ❙✐♥❝❡ (Tt) ❛♥❞ (St) ❛r❡ ✐s♦♠♦r♣❤✐❝ ✇❡❤❛✈❡ ψTt = Stψ ❢♦r t ∈ R✳ ▲❡t t0 ∈ R ❜❡ s✉❝❤ t❤❛t (Tt) ❤❛s t❤❡ sR(t0, {−1, 1}) ♣r♦♣❡rt② ❛♥❞ s✉❝❤ t❤❛tTt0 ❛♥❞ St0 ❛r❡ ❡r❣♦❞✐❝✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t (St) ❛❧s♦ ❤❛s t❤❡ sR(t0, {−1, 1}) ♣r♦♣❡rt②✳ ❋♦r s✐♠♣❧✐❝✐t② ♦❢♥♦t❛t✐♦♥ ❛ss✉♠❡ t❤❛t t0 = 1 ✭✇❡ ❤❛✈❡ ψT = Sψ✮✳ ❋✐① ǫ > 0 ❛♥❞ N ∈ N✳ ❇② ❊❣♦r♦✈✬s t❤❡♦r❡♠ t❤❡r❡❡①✐sts ❛ s❡t Bψ ⊂ X µ(Bψ) > 1− ǫ3 ❛♥❞ ǫ′ = ǫ′(ǫ) > 0 s✉❝❤ t❤❛t

✭❆✳✶✮ ❢♦r ❡✈❡r② x, y ∈ Bψ, dT (x, y) < ǫ′ ✇❡ ❤❛✈❡ dS(ψx, ψy) < ǫ2.

▲❡t κ = κ(S) = κ(T )(ǫ′) ✭κ(T ) ❝♦♠✐♥❣ ❢r♦♠ ❙❘✲♣r♦♣❡rt② ❢♦r T ✇✐t❤ ǫ′✮✳ ❇② ▲✉③✐♥✬s ❧❡♠♠❛✱ t❤❡r❡❡①✐sts N0 ∈ N ❛♥❞ ❛ s❡t CT ∈ X✱ µ(CT ) ≥ 1− ǫ2 s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ∈ CT ❛♥❞ M,L ≥ N0✱

LM ≥ κ

✭❆✳✷✮1

L

M+L∑

i=M

χBψ(Tix) ≥ 1− ǫ2.

❉❡♥♦t❡ N = max(N,N0)✳ ▲❡t ZT = ZT (ǫ′, N)✱ µ(ZT ) ≥ 1−ǫ′ ❛♥❞ δT = δT (ǫ

′, N) ❜❡ t❤❡ ❙❘✲♣❛r❛♠❡t❡rs

❢♦r ǫ′ ❛♥❞ N ✳ ❯s✐♥❣ ❊❣♦r♦✈✬s t❤❡♦r❡♠✱ t❤❡r❡ ❡①✐sts ❛ s❡t Vψ✱ ν(Vψ) ≥ 1− ǫ3 ❛♥❞ δ′ = δ′(δT ) s✉❝❤ t❤❛t

✭❆✳✸✮ ❢♦r ❡✈❡r② x, y ∈ Vψ, dS(x, y) < δ′ ✇❡ ❤❛✈❡ dS(ψ−1x, ψ−1y) < δT .

❉❡✜♥❡ δS = δS(ǫ,N) = δ′ ❛♥❞

ZS = ZS(ǫ,N) + ψ(ZT ∩ CT ) ∩ Vψ.

◆♦t✐❝❡ t❤❛t ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ZT , CT ❛♥❞ Vψ ✇❡ ❣❡t t❤❛t ν(ZS) ≥ 1− ǫ✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t ZS ❛♥❞ δSs❛t✐s❢② t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❙❘✲♣r♦♣❡rt②✳ ❋♦r t❤✐s ❛✐♠ ❧❡t✬s t❛❦❡ x, y ∈ ZS s✉❝❤ t❤❛t dS(x, y) < δS ✳ ❚❤❡♥

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✹✶

❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ZS ❛♥❞ ✭❆✳✸✮ ✐t ❢♦❧❧♦✇s t❤❛t ψ−1x, ψ−1y ∈ ZT ∩CT ❛♥❞ dT (ψ−1x, ψ−1y) < δT ✳ ❇②

t❤❡ ❙❘✲♣r♦♣❡rt② ❢♦r T ✐t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ❡①✐st M,L ≥ N ✱ L/M ≥ κ ❛♥❞ p ∈ {−1, 1} s✉❝❤ t❤❛t

✭❆✳✹✮1

L{i ∈ [M,M + L] : dT (T

i(ψ−1x), T i+p(ψ−1y)) < ǫ′} > 1− ǫ2.

❇✉t s✐♥❝❡ ψ−1x, ψ−1y ∈ CT ✐t ❢♦❧❧♦✇s t❤❛t

1

L{i ∈ [M,M + L] : T i(ψ−1x), T i+p(ψ−1y) ∈ Bψ} > 1− ǫ2.

❚❤❡r❡❢♦r❡✱ ❜② ✭❆✳✶✮

✭❆✳✺✮1

L{i ∈ [M,M + L] : dS(ψ(T

i(ψ−1x)), ψ(T i+p(ψ−1y))) < ǫ} > 1− ǫ2.

❇② ✭❆✳✹✮✱ ✭❆✳✺✮ ❛♥❞ t❤❡ ❢❛❝t t❤❛t ψT iψ−1 = Si ❢♦r i ∈ Z ✇❡ ❣❡t

1

L{i ∈ [M,M + L] : dS(S

ix, Si+py) < ǫ} > 1− ǫ.

❚❤❡r❡❢♦r❡ (St) ✐♥❞❡❡❞ ❤❛s t❤❡ sR(1, {1,−1}) ♣r♦♣❡rt②✳ ❚❤✐s ✜♥✐s❤❡s t❤❡ ♣r♦♦❢✳�

❆✳✷✳ ❙✐♥❣✉❧❛r✐t✐❡s ❞✐st❛♥❝❡s ❝♦♥tr♦❧ ❜② ♣♦s✐t✐✈❡ ❘❛✉③②✲❱❡❡❝❤ t✐♠❡s✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✐♥❝❧✉❞❡t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✺✳✸ ✉s❡❞ ✐♥ ❙❡❝t✐♦♥ ✺✳✶✳ ❚❤❡ ♣r♦♦❢ ✐s ❛ ♠✐♥♦r ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛❈✳✶ ❛♥❞ ❈♦r♦❧❧❛r② ❈✳✷ ✐♥ t❤❡ ♣❛♣❡r ❬✶✹❪ ❜② ❍✉❜❡rt✱ ▼❛r❝❤❡s❡ ❛♥❞ t❤❡ t❤✐r❞ ❛✉t❤♦r✱ r❡✇r✐tt❡♥ ✇✐t❤ t❤❡♥♦t❛t✐♦♥ ✉s❡❞ ✐♥ t❤✐s ♣❛♣❡r ❢♦r ❝♦♥✈❡♥✐❡♥❝❡ ♦❢ t❤❡ r❡❛❞❡r✳

❋♦r t❤❡ r❡st ♦❢ t❤❡ s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t T ✐s ❛♥ ■❊❚ ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❑❡❛♥❡ ❝♦♥❞✐t✐♦♥ ❛♥❞t❤❛t {nℓ}ℓ∈N ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ν✲❜❛❧❛♥❝❡❞ ✐♥❞✉❝t✐♦♥ t✐♠❡s ❢♦r T s✉❝❤ t❤❛t {nℓk}k∈N ✐s ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡

♦❢ t✐♠❡s ❢♦r s♦♠❡ ℓ ∈ N ✭❛s ✐♥ t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ Pr♦♣♦s✐t✐♦♥ ✺✳✶ ❛♥❞ ▲❡♠♠❛ ✺✳✸✮✳ ▲❡t ✉s r❡♠❛r❦ t❤❛t✱✉s✐♥❣ t❤❡ ♥♦t❛t✐♦♥ ✐♥tr♦❞✉❝❡❞ ✐♥ ✭✺✳✹✮ ❛♥❞ ✭✺✳✻✮✱ t❤❡ s❡ts

Dℓ +

{l(nℓ)α,t , α ∈ A\{α

(nℓ)1,t }

}={r(nℓ)α,t , α ∈ A\{α

(nℓ)d,t }

},

D−1ℓ +

{l(nℓ)α,b , α ∈ A\{α

(nℓ)1,b }

}={r(nℓ)α,b , α ∈ A\{α

(nℓ)d,b }

}

❝♦♥s✐st r❡s♣❡❝t✐✈❡❧② ♦❢ t❤❡ ❞✐s❝♦♥t✐♥✉t✐❡s ♦❢ T (nℓ) ❛♥❞ ✐ts ✐♥✈❡rs❡ (T (nℓ))−1✳ ❘❡❝❛❧❧ t❤❛t ✇❡ ✇r✐t❡ B > 0✐❢ ❛❧❧ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡ ♠❛tr✐① B ❛r❡ str✐❝t❧② ♣♦s✐t✐✈❡✳

❚♦ ✉♥❞❡rst❛♥❞ t❤❡ ❞❡t❛✐❧s ♦❢ t❤❡ ♣r♦♦❢✱ ✐t ✐s ✉s❡❢✉❧ t♦ ❦❡❡♣ ✐♥ ♠✐♥❞ t❤❡ ♠❛✐♥ ✐❞❡❛ ❜❡❤✐♥❞ ✐t✱ ✇❤✐❝❤ ✐s❜❛s❡❞ ♦♥ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❡✛❡❝t ♦❢ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥✿ t❤❡ (n+1)th st❡♣s ♦❢ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥

T (n+1) ✐s ♦❜t❛✐♥❡❞ ❜② ✐♥❞✉❝✐♥❣ T (n) ♦♥ ❛♥ ✐♥t❡r✈❛❧ I(n+1) ✇❤♦s❡ r✐❣❤t ❡♥❞♣♦✐♥t ✐s t❤❡ ❞✐s❝♦♥t✐♥✉✐t② ♦❢❡✐t❤❡r T (n) ♦r ✐ts ✐♥✈❡rs❡ ✇❤✐❝❤ ✐s ❝❧♦s❡st t♦ t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♦❢ I(n)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤✐s ✐♠♣❧✐❡s t❤❛tt❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❞✐s❝♦♥t✐♥✉✐t✐❡s ✭t❤❡ ❡♥❞♣♦✐♥t ♦❢ I(n)✱ ✇❤✐❝❤ ✐s ❛ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ T (n−1)✱ ❛♥❞

t❤❡ ❝❧♦s❡st ❞✐s❝♦♥t✐♥✉✐t② ♦❢ T (n)✮ ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② t❤❡ ❧❡♥❣❤t ♦❢ ❛♥ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡❞ ❜② T (n+1)✳

❚❤✉s✱ st❛rt✐♥❣ ❢r♦♠ T (nℓ)✱ s✐♥❝❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ T (nℓ) ✭❛♥❞ ✐ts ✐♥✈❡rs❡✮ ❛♣♣❡❛r ❛s ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢

T (n) ✭❛♥❞ ✐ts ✐♥✈❡rs❡✮ ❢♦r n ≥ nℓ✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥❞✉❝t✐♦♥ st❡♣s ✉♣ t♦ t❤❡ ♥❡①t ❜❛❧❛♥❝❡❞ st❡♣ T (nℓ+1)

❣✉❛r❛♥t❡❡s t❤❛t t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛❧❧ ♣❛✐rs ♦❢ ❞✐s❝♦♥t✐♥✉✐t✐❡s ✐♥ Dℓ ❛♥❞ D−1ℓ ❝❛♥ ❜❡ ❝♦♥tr♦❧❧❡❞ ❜②

❧❡♥❣❤ts ♦❢ ❛♥ ✐♥t❡r✈❛❧ ♦❢ s♦♠❡ T (n) ✇✐t❤ nℓ ≤ n ≤ nℓ+1✱ ❛♥❞ ❤❡♥❝❡ ✭❜② ♠♦♥♦t♦♥✐❝✐t② ❛♥❞ ❜❛❧❛♥❝❡✮ ✐♥

t❡r♠s ♦❢ t❤❡ ❧❡♥❣❤t ♦❢ I(nℓ+1)✳

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✺✳✸✳ ▲❡t ✉s ✜rst s❤♦✇ t❤❛t✱ s✐♥❝❡ B(nℓ,nℓ+1) > 0✱

✭❆✳✻✮[0, λ(nℓ+1)

)∩(Dℓ ∪D

−1ℓ

)= ∅

✭❝♦♠♣❛r❡ ✇✐t❤ ▲❡♠♠❛ ❈✳✶ ✐♥ ❬✶✹❪✮✳ ❘❡❝❛❧❧ ✭s❡❡ ✭✷✳✸✮ ✐♥ ❙❡❝t✐♦♥ ✷✳✺ ❛♥❞ t❤❡ ♥♦t❛t✐♦♥ t❤❡r❡❛❢t❡r✮ t❤❛t

✇❡ ❤❛✈❡ λ(nℓ)α =

∑χ∈AB

(nℓ,nℓ+1)αχ λ

(nℓ+1)χ ❢♦r ❛♥② ❧❡tt❡r α ∈ A✳ ❚❤❡r❡❢♦r❡✱ s✐♥❝❡ ❛❧❧ t❤❡ ❡♥tr✐❡s ♦❢ t❤❡

♠❛tr✐① B(nℓ,nℓ+1) ❛r❡ ♣♦s✐t✐✈❡ ❛♥❞ ❤❡♥❝❡✱ ❜❡✐♥❣ ✐♥t❡❣❡rs✱ ❛r❡ ❣r❡❛t❡r t❤❛♥ 1✱ ✇❡ ❤❛✈❡ t❤❛t minα∈A λ(nℓ)α ≥∑

χ∈A λ(nℓ+1)χ = λ(nℓ+1)✳ ❚❤✉s✱ ✭❆✳✻✮ ❢♦❧❧♦✇s s✐♥❝❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ Dℓ ♦r D

−1ℓ ❛r❡ ❛❧❧ r✐❣❤t ❡♥❞♣♦✐♥ts ♦❢

✹✷ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

✉♥✐♦♥ ♦❢ ✐♥t❡r✈❛❧s ✇❤♦s❡ ❧❡♥❣t❤s ❛❧❧ ❜❡❧♦♥❣ t♦ t❤❡ s❡t {λ(nℓ)χ ;χ ∈ A} ❛♥❞ ❤❡♥❝❡ ❡❛❝❤ ♦❢ t❤❡♠ ✐s ❣r❡❛t❡r

t❤❛♥ ❛ ♥♦♥ tr✐✈✐❛❧ s✉♠ ♦❢ t❤❡s❡ ❧❡♥❣❤ts✳

❲❡ ❝❛♥ ♥♦✇ ✜♥✐s❤ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✺✳✸✳ ❆ss✉♠❡ ✜rst t❤❛t α = α(nℓ)t,1 ✱ s♦ t❤❛t l

(nℓ)α,t = 0✳ ■♥

t❤✐s ❝❛s❡✱ s✐♥❝❡ ✇❡ ❛r❡ ❛ss✉♠✐♥❣ t❤❛t β 6= α(nℓ)1,b ❛♥❞ ❤❡♥❝❡ l

(nℓ)β,b 6= 0✱ ✉s✐♥❣ t❤❛t nℓ ✐s ν✲❜❛❧❛♥❝❡❞ ❜②

❛ss✉♠♣t✐♦♥ ✭r❡❝❛❧❧ ❘❡♠❛r❦ ✷✳✶✶✮✱ ✇❡ ❤❛✈❡ t❤❛t |l(nℓ)β,b | ≥ minχ λ

(nℓ)χ ≥ 1

νλ(nℓ) ≥ 1

νλ(nℓ+ℓ) ❛♥❞ ❤❡♥❝❡ ✭✺✳✼✮

❤♦❧❞s tr✐✈✐❛❧❧② ✐♥ t❤✐s ❝❛s❡✳ ❆ss✉♠❡ ♥❡①t t❤❛t α 6= α(nℓ)1,t ❛♥❞ β 6= α

(nℓ)1,b ✱ s♦ t❤❛t α ∈ Dℓ ❛♥❞ β ∈ D−1

ℓ ✳

❈♦♥s✐❞❡r t❤❡ ♠✐♥✐♠✉♠ n ≥ nℓ s✉❝❤ t❤❛t ❜♦t❤ l(nℓ)α,t ❛♥❞ l

(nℓ)β,b ❞♦ ♥♦t ❜❡❧♦♥❣ t♦ t❤❡ ✐♥t❡r✐♦r ♦❢ I(n)✳ ❇②

✭❆✳✻✮✱ n ≤ nℓ+1✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ ❘❛✉③②✲❱❡❡❝❤ ✐♥❞✉❝t✐♦♥ ❛♥❞ n✱ ✐❢ l(nℓ)α,t > l

(nℓ)β,b ✱ l

(nℓ)α,t ✐s t❤❡ ❝❧♦s❡st

❞✐s❝♦♥t✐♥✉✐t② ♦❢ T (n−1) t♦ t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♦❢ I(n−1) ❛♥❞ I(n) = [0, l(nℓ)α,t )✱ ♦r✱ ✐❢ l

(nℓ)α,t < l

(nℓ)β,b ✱ t❤❡♥ l

(nℓ)β,b

✐s t❤❡ ❝❧♦s❡st ❞✐s❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ T (n−1) t♦ t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ♦❢ I(n−1) ❛♥❞ I(n) = [0, l(nℓ)β,b )✳

■♥ t❤❡ ✜rst ❝❛s❡✱ I(n) ✐s ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ❢r♦♠ I(n−1) ❛♥ ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣❤t λ(n−1)α ✭s✐♥❝❡ ✐♥ t❤✐s ❝❛s❡

α = α(n−1)t,d ✮✱ ✇❤✐❧❡ ✐♥ t❤❡ s❡❝♦♥❞ ♦❢ ❧❡♥❣t❤ λ

(n−1)β ✭s✐♥❝❡ ✐♥ t❤❛t ❝❛s❡ β = α

(n−1)b,d ✮✳ ■♥ ❜♦t❤ ❝❛s❡s✱ ✉s✐♥❣

t❤❛t ❢♦r ❛♥② χ ∈ A t❤❡ s❡q✉❡♥❝❡ ♦❢ ❧❡♥❣❤ts (λ(k)α )k ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ✐♥ k ❛♥❞ r❡❝❛❧❧✐♥❣ t❤❛t t❤❡ st❡♣

nℓ+1 ✐s ν✲❜❛❧❛♥❝❡❞ ❜② ❛ss✉♠♣t✐♦♥ ✭r❡❝❛❧❧ ❘❡♠❛r❦ ✷✳✶✶✮✱ ✇❡ ❤❛✈❡ t❤❛t

|l(nℓ)α,t − l

(nℓ)β,b | ≥ min

χ∈Aλ(n−1)χ ≥ min

χ∈Aλ(nℓ+1)χ ≥

1

νλ(nℓ+1).

❚❤✐s ❝♦♥❝❧✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ ✭✺✳✼✮✳ ❚♦ ♣r♦✈❡ ✭✺✳✽✮✱ ✐t ✐s ❡♥♦✉❣❤ t♦ r❡♠❛r❦ t❤❛t ✐❢ β 6= α(nℓ)d,b ✱ s✐♥❝❡ ❜②

❛ss✉♠♣t✐♦♥ α 6= α(nℓ+1)d,t ✱ ✭✺✳✽✮ r❡❞✉❝❡s t♦ ✭✺✳✼✮ ❜② ❘❡♠❛r❦ ✺✳✷✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ β = β

(nℓ)d,b ✇❡ ❤❛✈❡

t❤❛t r(nℓ)β,b = 1✱ ❛♥❞✱ ❜② t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t α 6= α

(nℓ+1)d,t ✱ r

(nℓ)α,t ✐s ♥♦t t❤❡ ❡♥❞♣♦✐♥t ♦❢ t❤❡ ❧❛st ✐♥t❡r✈❛❧

❡①❝❤❛♥❣❡❞ ❜② T (nℓ)✳ ❚❤✉s✱ ✉s✐♥❣ ❛❣❛✐♥ ν✲❜❛❧❛♥❝❡ ♦❢ nℓ✱ ✇❡ ❤❛✈❡ t❤❛t |r(nℓ)α,t −1| ≥ minχ λ

(nℓ)χ ≥ 1

νλ(nℓ) ≥

1νλ

(nℓ+ℓ). ❚❤✐s ❝♦♥❝❧✉❞❡s t❤❡ ♣r♦♦❢ ♦❢ ✭✺✳✽✮ ❛♥❞ ❤❡♥❝❡ ♦❢ t❤❡ ▲❡♠♠❛✳ �

❆❝❦♥♦✇❧❡❞❣♠❡♥ts

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ▼✳ ▲❡♠❛➠❝③②❦ ❛♥❞ ❏✳P✳ ❚❤♦✉✈❡♥♦t ❢♦r t❤❡✐r ✐♥t❡r❡st ✐♥ t❤❡ q✉❡st✐♦♥s ❤❡r❡❛❞❞r❡ss❡❞✳ ❏✳P✳ ❚❤♦✉✈❡♥♦t ❛s❦❡❞ ❈✳❯✳ s❡✈❡r❛❧ ②❡❛rs ❛❣♦ ✇❤❡t❤❡r t❤❡ ✢♦✇s s❤❡ ♣r♦✈❡❞ t♦ ❜❡ ♠✐①✐♥❣ ✐♥❬✹✶❪ ❛r❡ ♠✐①✐♥❣ ♦❢ ❛❧❧ ♦r❞❡rs ❛♥❞ s✉❣❣❡st❡❞ t♦ tr② t♦ ♣r♦✈❡ t❤❡ ❘❛t♥❡r ♣r♦♣❡rt② ❢♦r t❤❡♠❀ ▼✳ ▲❡♠❛➠❝③②❦❤❛s ✐♥s♣✐r❡❞ ❛♥❞ ♠♦t✐✈❛t❡❞ t❤❡ ❛✉t❤♦rs✱ ✐♥ ♣❛rt✐❝✉❧❛r ❆✳❑✳✱ t♦ ❧♦♦❦ ❢♦r s✉✐t❛❜❧❡ ✈❛r✐❛t✐♦♥s ♦❢ t❤❡ ❘❛t♥❡r♣r♦♣❡rt②✳ ❲❡ ❛❧s♦ t❤❛♥❦ ❤✐♠ ❢♦r ✉s❡❢✉❧ ❞✐s❝✉ss✐♦♥s✳ ❲❡ ❛r❡ t❤❛♥❦❢✉❧ t♦ ❏✳ ❈❤❛✐❦❛ ❢♦r ❤✐s ❝♦♠♠❡♥ts ♦♥ t❤❡✜rst ✈❡rs✐♦♥ ♦❢ t❤✐s ♣❛♣❡r ❛♥❞ t♦ t❤❡ r❡❢❡r❡❡ ♦❢ t❤❡ ♣❛♣❡r ❢♦r ❤✐s✴❤❡r ❝❛r❡❢✉❧ r❡❛❞✐♥❣ ❛♥❞ ❝♦rr❡❝t✐♦♥s✳ ❚❤❡❝♦❧❧❛❜♦r❛t✐♦♥ t❤❛t ❧❡❞ t♦ t❤✐s ♣❛♣❡r ✇❛s st❛rt❡❞ ✐♥ ♦❝❝❛s✐♦♥ ♦❢ t❤❡ ❊r❣♦❞✐❝ ❚❤❡♦r② ❛♥❞ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s❝♦♥❢❡r❡♥❝❡ ❤❡❧❞ ✐♥ ❚♦r✉➠ ✐♥ ▼❛② ✷✵✶✹❀ ✇❡ t❤❛♥❦ t❤❡ ♦r❣❛♥✐③❡rs ❛♥❞ t❤❡ ❢✉♥❞✐♥❣ ❜♦❞✐❡s ❢♦r ♣r♦✈✐❞✐♥❣✉s t❤❡ ♦♣♣♦rt✉♥✐t② t♦ ❜❡❣✐♥ t❤✐s ✇♦r❦✳ ❏✳ ❑✳✲P✳ ✐s s✉♣♣♦rt❡❞ ❜② ◆❛r♦❞♦✇❡ ❈❡♥tr✉♠ ◆❛✉❦✐ ❣r❛♥t❯▼❖✲✷✵✶✹✴✶✺✴❇✴❙❚✶✴✵✸✼✸✻❀ ❈✳ ❯✳ ✐s s✉♣♣♦rt❡❞ ❜② t❤❡ ❊❘❈ ❣r❛♥t ❈❤❛P❛r❉②♥ ❛♥❞ ❜② t❤❡ ▲❡✈❡r❤✉❧♠❚r✉st t❤r♦✉❣❤ ❛ ▲❡✈❡r❤✉❧♠❡ Pr✐③❡✳ ❚❤❡ r❡s❡❛r❝❤ ❧❡❛❞✐♥❣ t♦ t❤❡s❡ r❡s✉❧ts ❤❛s r❡❝❡✐✈❡❞ ❢✉♥❞✐♥❣ ❢r♦♠ t❤❡❊✉r♦♣❡❛♥ ❘❡s❡❛r❝❤ ❈♦✉♥❝✐❧ ✉♥❞❡r t❤❡ ❊✉r♦♣❡❛♥ ❯♥✐♦♥ ❙❡✈❡♥t❤ ❋r❛♠❡✇♦r❦ Pr♦❣r❛♠♠❡ ✭❋P✴✷✵✵✼✲✷✵✶✸✮✴ ❊❘❈ ●r❛♥t ❆❣r❡❡♠❡♥t ♥✳ ✸✸✺✾✽✾✳

❘❡❢❡r❡♥❝❡s

❬✶❪ ❱✳ ■✳ ❆r♥♦❧❞✱ ❚♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❡r❣♦❞✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❝❧♦s❡❞ ✶✲❢♦r♠s ✇✐t❤ ✐♥❝♦♠♠❡♥s✉r❛❜❧❡ ♣❡r✐♦❞s✳✱ ❋✉♥❦ts✐♦♥❛❧✬♥②✐❆♥❛❧✐③ ✐ ❊❣♦ Pr✐❧♦③❤❡♥✐②❛✱ ✷✺ ✭✶✾✾✶✮✱ ♣♣✳ ✶✕✶✷✳ ✭❚r❛♥s❧❛t❡❞ ✐♥✿ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✺✿✷✿✽✶✕✾✵✱✶✾✾✶✮✳

❬✷❪ ❆✳ ❆✈✐❧❛ ❛♥❞ ●✳ ❋♦r♥✐✱ ❲❡❛❦ ♠✐①✐♥❣ ❢♦r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ tr❛♥s❧❛t✐♦♥ ✢♦✇s✱ ❆♥♥✳ ♦❢ ▼❛t❤✳✭✷✮✱ ✶✻✺ ✭✷✵✵✼✮✱ ♣♣✳ ✻✸✼✕✻✻✹✳

❬✸❪ ❆✳ ❆✈✐❧❛✱ ❙✳ ●♦✉ë③❡❧✱ ❛♥❞ ❏✳✲❈✳ ❨♦❝❝♦③✱ ❊①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ❢♦r t❤❡ ❚❡✐❝❤♠ü❧❧❡r ✢♦✇✱ P✉❜❧✳ ▼❛t❤✳ ■♥st✳ ❍❛✉t❡s➱t✉❞❡s ❙❝✐✳✱ ✭✷✵✵✻✮✱ ♣♣✳ ✶✹✸✕✷✶✶✳

❬✹❪ ❆✳ ❆✈✐❧❛ ❛♥❞ ▼✳ ❱✐❛♥❛✱ ❙✐♠♣❧✐❝✐t② ♦❢ ▲②❛♣✉♥♦✈ s♣❡❝tr❛✿ ♣r♦♦❢ ♦❢ t❤❡ ❩♦r✐❝❤✲❑♦♥ts❡✈✐❝❤ ❝♦♥❥❡❝t✉r❡✱ ❆❝t❛ ▼❛t❤✳✱✶✾✽ ✭✷✵✵✼✮✱ ♣♣✳ ✶✕✺✻✳

▼❯▲❚■P▲❊ ▼■❳■◆● ■◆ ❆❘❊❆ P❘❊❙❊❘❱■◆● ❋▲❖❲❙ ✹✸

❬✺❪ ❆✳ ■✳ ❇✉❢❡t♦✈✱ ❉❡❝❛② ♦❢ ❝♦rr❡❧❛t✐♦♥s ❢♦r t❤❡ ❘❛✉③②✲❱❡❡❝❤✲❩♦r✐❝❤ ✐♥❞✉❝t✐♦♥ ♠❛♣ ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ t❤❡ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r t❤❡ ❚❡✐❝❤♠ü❧❧❡r ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡ ♦❢ ❛❜❡❧✐❛♥ ❞✐✛❡r❡♥t✐❛❧s✱❏✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✾ ✭✷✵✵✻✮✱ ♣♣✳ ✺✼✾✕✻✷✸✳

❬✻❪ ✱ ▲✐♠✐t t❤❡♦r❡♠s ❢♦r tr❛♥s❧❛t✐♦♥ ✢♦✇s✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✭✷✮✱ ✶✼✾ ✭✷✵✶✹✮✱ ♣♣✳ ✹✸✶✕✹✾✾✳❬✼❪ ❏✳ ❈❤❛✐❦❛✱ ❊✈❡r② ❡r❣♦❞✐❝ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❞✐s❥♦✐♥t ❢r♦♠ ❛❧♠♦st ❡✈❡r② ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ❆♥♥✳ ♦❢

▼❛t❤✳ ✭✷✮✱ ✶✼✺ ✭✷✵✶✷✮✱ ♣♣✳ ✷✸✼✕✷✺✸✳❬✽❪ ❏✳ ❈❤❛✐❦❛ ❛♥❞ ❆✳ ❲r✐❣❤t✱ ❆ s♠♦♦t❤ ♠✐①✐♥❣ ✢♦✇ ♦♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts✱ ✭✷✵✶✺✮✳❬✾❪ ■✳ P✳ ❈♦r♥❢❡❧❞✱ ❙✳ ❱✳ ❋♦♠✐♥✱ ❛♥❞ ❨✳ ●✳ ❙✐♥❛✐✱ ❊r❣♦❞✐❝ ❚❤❡♦r②✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✶✾✽✵✳❬✶✵❪ ❇✳ ❋❛②❛❞ ❛♥❞ ❆✳ ❑❛♥✐❣♦✇s❦✐✱ ▼✉❧t✐♣❧❡ ♠✐①✐♥❣ ❢♦r ❛ ❝❧❛ss ♦❢ ❝♦♥s❡r✈❛t✐✈❡ s✉r❢❛❝❡ ✢♦✇s✱ ■♥✈❡♥t✐♦♥❡s ♠❛t❤❡♠❛t✐❝❛❡✱

✭✷✵✶✺✮✱ ♣♣✳ ✶✕✻✵✳ P✉❜❧✐s❤❡❞ ♦♥❧✐♥❡✳❬✶✶❪ ❑✳ ❋r❛☛❝③❡❦ ❛♥❞ ▼✳ ▲❡♠❛➠❝③②❦✱ ❖♥ ♠✐❧❞ ♠✐①✐♥❣ ♦❢ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s ✉♥❞❡r ♣✐❡❝❡✇✐s❡ s♠♦♦t❤

❢✉♥❝t✐♦♥s✱ ❊r❣♦❞✐❝ ❚❤❡♦r② ❛♥❞ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✱ ✷✻ ✭✷✵✵✻✮✱ ♣♣✳ ✼✶✾✕✼✸✽✳❬✶✷❪ ✱ ❘❛t♥❡r✬s ♣r♦♣❡rt② ❛♥❞ ♠✐❧❞ ♠✐①✐♥❣ ❢♦r s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r t✇♦✲❞✐♠❡♥s✐♦♥❛❧ r♦t❛t✐♦♥s✱ ❏✳ ▼♦❞✳ ❉②♥✳✱ ✹ ✭✷✵✶✵✮✱

♣♣✳ ✻✵✾✕✻✸✺✳❬✶✸❪ ❊✳ ●❧❛s♥❡r✱ ❊r❣♦❞✐❝ t❤❡♦r② ✈✐❛ ❥♦✐♥✐♥❣s✱ ✈♦❧✳ ✶✵✶ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙✉r✈❡②s ❛♥❞ ▼♦♥♦❣r❛♣❤s✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧

❙♦❝✐❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✷✵✵✸✳❬✶✹❪ P✳ ❍✉❜❡rt✱ ▲✳ ▼❛r❝❤❡s❡✱ ❛♥❞ ❈✳ ❯❧❝✐❣r❛✐✱ ▲❛❣r❛♥❣❡ s♣❡❝tr❛ ✐♥ ❚❡✐❝❤♠ü❧❧❡r ❞②♥❛♠✐❝s ✈✐❛ r❡♥♦r♠❛❧✐③❛t✐♦♥✱

●❡♦♠✳ ❋✉♥❝t✳ ❆♥❛❧✳✱ ✷✺ ✭✷✵✶✺✮✱ ♣♣✳ ✶✽✵✕✷✺✺✳❬✶✺❪ ❆✳ ❑❛♥✐❣♦✇s❦✐ ❛♥❞ ❏✳ ❑✉➟❛❣❛✲Pr③②♠✉s✱ ❘❛t♥❡r✬s ♣r♦♣❡rt② ❛♥❞ ♠✐❧❞ ♠✐①✐♥❣ ❢♦r s♠♦♦t❤ ✢♦✇s ♦♥ s✉r❢❛❝❡s✱ ❊r❣♦❞✐❝

❚❤❡♦r② ❉②♥❛♠✳ ❙②st❡♠s✱ ✸✻ ✭✷✵✶✻✮✱ ♣♣✳ ✷✺✶✷✕✷✺✸✼✳❬✶✻❪ ❆✳ ❇✳ ❑❛t♦❦✱ ■♥✈❛r✐❛♥t ♠❡❛s✉r❡s ♦❢ ✢♦✇s ♦♥ ♦r✐❡♥t❡❞ s✉r❢❛❝❡s✳✱ ❙♦✈✐❡t ▼❛t❤❡♠❛t✐❝s✳ ❉♦❦❧❛❞②✱ ✶✹ ✭✶✾✼✸✮✱ ♣♣✳ ✶✶✵✹✕

✶✶✵✽✳❬✶✼❪ ▼✳ ❑❡❛♥❡✱ ■♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛s❢♦r♠❛t✐♦♥s✱ ▼❛t❤❡♠❛t✐s❝❤❡ ❩❡✐ts❝❤r✐❢t✱ ✶✹✶ ✭✶✾✼✺✮✱ ♣♣✳ ✷✺✕✸✶✳❬✶✽❪ ❙✳ P✳ ❑❡r❝❦❤♦❢❢✱ ❙✐♠♣❧✐❝✐❛❧ s②st❡♠s ❢♦r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ ♠❛♣s ❛♥❞ ♠❡❛s✉r❡❞ ❢♦❧✐❛t✐♦♥s✳✱ ❊r❣♦❞✐❝ ❚❤❡♦r② ❛♥❞

❉②♥❛♠✐❝❛❧ ❙②st❡♠s✱ ✺ ✭✶✾✽✺✮✱ ♣♣✳ ✷✺✼✕✷✼✶✳❬✶✾❪ ❆✳ ❱✳ ❑♦↔❡r❣✐♥✱ ❚❤❡ ❛❜s❡♥❝❡ ♦❢ ♠✐①✐♥❣ ✐♥ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ❛ r♦t❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ❛♥❞ ✐♥ ✢♦✇s ♦♥ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧

t♦r✉s✳✱ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦ ❙❙❙❘✱ ✷✵✺ ✭✶✾✼✷✮✱ ♣♣✳ ✺✶✷✕✺✶✽✳ ✭❚r❛♥s❧❛t❡❞ ✐♥✿ ❙♦✈✐❡t ▼❛t❤✳ ❉♦❦❧✳✱ ✶✸✿✾✹✾✲✾✺✷✱ ✶✾✼✷✮✳❬✷✵❪ ✱ ▼✐①✐♥❣ ✐♥ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r ❛ s❤✐❢t✐♥❣ ♦❢ s❡❣♠❡♥ts ❛♥❞ ✐♥ s♠♦♦t❤ ✢♦✇s ♦♥ s✉r❢❛❝❡s✳✱ ▼❛t✳ ❙❜✳✱ ✾✻ ✭✶✾✼✺✮✱

♣♣✳ ✹✼✶✕✺✵✷✳❬✷✶❪ ❆✳ ❱✳ ❑♦❝❤❡r❣✐♥✱ ◆♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts ❛♥❞ ♠✐①✐♥❣ ✐♥ ✢♦✇s ♦♥ ❛ ✷✲t♦r✉s✳✱ ▼❛t❡♠❛t✐❝❤❡s❦✐✐ ❙❜♦r♥✐❦✱ ✶✾✹

✭✷✵✵✸✮✱ ♣♣✳ ✽✸✕✶✶✷✳ ✭❚r❛♥s❧❛t❡❞ ✐♥✿ ❙❜✳ ▼❛t❤✳✱ ✶✾✹✭✽✮✿✶✶✾✺✲✶✷✷✹✮✳❬✷✷❪ ✱ ◆♦♥✲❞❡❣❡♥❡r❛t❡ ✜①❡❞ ♣♦✐♥ts ❛♥❞ ♠✐①✐♥❣ ✐♥ ✢♦✇s ♦♥ ❛ ✷✲t♦r✉s✳■■✳✱ ▼❛t❡♠❛t✐❝❤❡s❦✐✐ ❙❜♦r♥✐❦✱ ✶✾✺ ✭✷✵✵✹✮✱ ♣♣✳ ✽✸✕

✶✶✷✳ ✭❚r❛♥s❧❛t❡❞ ✐♥✿ ❙❜✳ ▼❛t❤✳✱ ✶✾✺✭✸✮✿✸✶✼✲✸✹✻✮✳❬✷✸❪ ✱ ❙♦♠❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡♦r❡♠s ♦♥ ♠✐①✐♥❣ ✢♦✇s ✇✐t❤ ♥♦♥❞❡❣❡♥❡r❛t❡ s❛❞❞❧❡s ♦♥ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ t♦r✉s✳✱

▼❛t✳ ❙❜✳✱ ✶✾✺ ✭✷✵✵✹✮✱ ♣♣✳ ✶✾✕✸✻✳❬✷✹❪ ✱ ❲❡❧❧✲❛♣♣r♦①✐♠❛❜❧❡ ❛♥❣❧❡s ❛♥❞ ♠✐①✐♥❣ ❢♦r ✢♦✇s ♦♥ T

2 ✇✐t❤ ♥♦♥s✐♥❣✉❧❛r ✜①❡❞ ♣♦✐♥ts✳✱ ❊❧❡❝tr♦♥✳ ❘❡s✳ ❆♥♥♦✉♥❝✳❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✵ ✭✷✵✵✹✮✱ ♣♣✳ ✶✶✸✕✶✷✶✳

❬✷✺❪ ●✳ ▲❡✈✐tt✱ ❋❡✉✐❧❧❡tt❛❣❡s ❞❡s s✉r❢❛❝❡s✱ t❤ès❡✱ ✶✾✽✸✳❬✷✻❪ ❙✳ ▼❛r♠✐✱ P✳ ▼♦✉ss❛✱ ❛♥❞ ❏✳✲❈✳ ❨♦❝❝♦③✱ ❚❤❡ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ❡q✉❛t✐♦♥ ❢♦r ❘♦t❤✲t②♣❡ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ ♠❛♣s✱ ❏✳

❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✽ ✭✷✵✵✺✮✱ ♣♣✳ ✽✷✸✕✽✼✷✳❬✷✼❪ ❙✳ ▼❛r♠✐✱ P✳ ▼♦✉ss❛✱ ❛♥❞ ❏✳✲❈✳ ❨♦❝❝♦③✱ ▲✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❣❡♥❡r❛❧✐③❡❞ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ ♠❛♣s✱ ❆♥♥✳ ♦❢ ▼❛t❤✳

✭✷✮✱ ✶✼✻ ✭✷✵✶✷✮✱ ♣♣✳ ✶✺✽✸✕✶✻✹✻✳❬✷✽❪ ❍✳ ▼❛s✉r✱ ■♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ ♠❡❛s✉r❡❞ ❢♦❧✐❛t✐♦♥s✱ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✶✺ ✭✶✾✽✷✮✱ ♣♣✳ ✶✻✾✕

✷✵✵✳❬✷✾❪ ❆✳ ▼❛②❡r✱ ❚r❛❥❡❝t♦r✐❡s ♦♥ t❤❡ ❝❧♦s❡❞ ♦r✐❡♥t❛❜❧❡ s✉r❢❛❝❡s✱ ❘❡❝✳ ▼❛t❤✳ ❬▼❛t✳ ❙❜♦r♥✐❦❪ ◆✳❙✳✱ ✶✷✭✺✹✮ ✭✶✾✹✸✮✱ ♣♣✳ ✼✶✕✽✹✳❬✸✵❪ ❚✳ ▼♦r✐t❛✱ ❘❡♥♦r♠❛❧✐③❡❞ ❘❛✉③② ✐♥❞✉❝t✐♦♥s✱ ✐♥ Pr♦❜❛❜✐❧✐t② ❛♥❞ ♥✉♠❜❡r t❤❡♦r②✖❑❛♥❛③❛✇❛ ✷✵✵✺✱ ✈♦❧✳ ✹✾ ♦❢ ❆❞✈✳

❙t✉❞✳ P✉r❡ ▼❛t❤✳✱ ▼❛t❤✳ ❙♦❝✳ ❏❛♣❛♥✱ ❚♦❦②♦✱ ✷✵✵✼✱ ♣♣✳ ✷✻✸✕✷✽✽✳❬✸✶❪ ■✳ ◆✐❦♦❧❛❡✈ ❛♥❞ ❊✳ ❩❤✉③❤♦♠❛✱ ❋❧♦✇s ♦♥ ✷✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞s✱ ✈♦❧✳ ✶✼✵✺ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱

❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ✶✾✾✾✳ ❆♥ ♦✈❡r✈✐❡✇✳❬✸✷❪ ❙✳ P✳ ◆♦✈✐❦♦✈✱ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r♠❛❧✐s♠ ❛♥❞ ❛ ♠✉❧t✐✈❛❧✉❡❞ ❛♥❛❧♦❣✉❡ ♦❢ ▼♦rs❡ t❤❡♦r②✱ ✭❘✉ss✐❛♥✮ ❯s♣❡❦❤✐ ▼❛t❡♠✲

❛t✐❝❤❡s❦✐❦❤ ◆❛✉❦✱ ✸✼ ✭✶✾✽✷✮✱ ♣♣✳ ✸✕✹✾✳ ✭❚r❛s❧❛t❡❞ ✐♥✿ ❘✉ss✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙✉r✈❡②s✱ ✸✼ ◆♦ ✺✿✶✕✺✻✱ ✶✾✽✷✮✳❬✸✸❪ ▼✳ ❘❛t♥❡r✱ ❍♦r♦❝②❝❧❡ ✢♦✇s✱ ❥♦✐♥✐♥❣s ❛♥❞ r✐❣✐❞✐t② ♦❢ ♣r♦❞✉❝ts✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✭✷✮✱ ✶✶✽ ✭✶✾✽✸✮✱ ♣♣✳ ✷✼✼✕✸✶✸✳❬✸✹❪ ●✳ ❘❛✉③②✱ ➱❝❤❛♥❣❡s ❞✬✐♥t❡r✈❛❧❧❡s ❡t tr❛s❢♦r♠❛t✐♦♥s ✐♥❞✉✐t❡s✱ ❆❝t❛ ❆r✐t❤♠❡t✐❝❛✱ ❳❳❳■❱ ✭✶✾✼✾✮✱ ♣♣✳ ✸✶✺✕✸✷✽✳❬✸✺❪ ❉✳ ❘❛✈♦tt✐✱ ◗✉❛♥t✐t❛t✐✈❡ ♠✐①✐♥❣ ❢♦r ❧♦❝❛❧❧② ❤❛♠✐❧t♦♥✐❛♥ ✢♦✇s ✇✐t❤ s❛❞❞❧❡ ❧♦♦♣s ♦♥ ❝♦♠♣❛❝t s✉r❢❛❝❡s✳ Pr❡♣r✐♥t

❛r❳✐✈✿✶✻✶✵✳✵✽✼✹✸✱ ✷✵✶✻✳❬✸✻❪ ❱✳ ❆✳ ❘♦❤❧✐♥✱ ❖♥ ❡♥❞♦♠♦r♣❤✐s♠s ♦❢ ❝♦♠♣❛❝t ❝♦♠♠✉t❛t✐✈❡ ❣r♦✉♣s✱ ■③✈❡st✐②❛ ❆❦❛❞✳ ◆❛✉❦ ❙❙❙❘✳ ❙❡r✳ ▼❛t✳✱ ✶✸ ✭✶✾✹✾✮✱

♣♣✳ ✸✷✾✕✸✹✵✳❬✸✼❪ ❱✳ ❱✳ ❘②③❤✐❦♦✈ ❛♥❞ ❏✳✲P✳ ❚❤♦✉✈❡♥♦t✱ ❉✐s❥♦✐♥t♥❡ss✱ ❞✐✈✐s✐❜✐❧✐t②✱ ❛♥❞ q✉❛s✐✲s✐♠♣❧✐❝✐t② ♦❢ ♠❡❛s✉r❡✲♣r❡s❡r✈✐♥❣ ❛❝✲

t✐♦♥s✱ ❋✉♥❦ts✐♦♥❛❧✳ ❆♥❛❧✳ ✐ Pr✐❧♦③❤❡♥✳✱ ✹✵ ✭✷✵✵✻✮✱ ♣♣✳ ✽✺✕✽✾✳❬✸✽❪ ❉✳ ❙❝❤❡❣❧♦✈✱ ❆❜s❡♥❝❡ ♦❢ ♠✐①✐♥❣ ❢♦r s♠♦♦t❤ ✢♦✇s ♦♥ ❣❡♥✉s t✇♦ s✉r❢❛❝❡s✱ ❏✳ ▼♦❞✳ ❉②♥✳✱ ✸ ✭✷✵✵✾✮✱ ♣♣✳ ✶✸✕✸✹✳

✹✹ ❆❉❆▼ ❑❆◆■●❖❲❙❑■✱ ❏❖❆◆◆❆ ❑❯❾❆●❆✲P❘❩❨▼❯❙✱ ❆◆❉ ❈❖❘■◆◆❆ ❯▲❈■●❘❆■

❬✸✾❪ ❨✳ ●✳ ❙✐♥❛✐ ❛♥❞ ❑✳ ▼✳ ❑❤❛♥✐♥✱▼✐①✐♥❣ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ s♣❡❝✐❛❧ ✢♦✇s ♦✈❡r r♦t❛t✐♦♥s ♦❢ t❤❡ ❝✐r❝❧❡✳✱ ❋✉♥❦ts✐♦♥❛❧✬♥②✐❆♥❛❧✐③ ✐ ❊❣♦ Pr✐❧♦③❤❡♥✐②❛✱ ✷✻ ✭✶✾✾✷✮✱ ♣♣✳ ✶✕✷✶✳ ✭❚r❛♥s❧❛t❡❞ ✐♥✿ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✻✿✸✿✶✺✺✕✶✻✾✱✶✾✾✷✮✳

❬✹✵❪ ❏✳✲P✳ ❚❤♦✉✈❡♥♦t✱ ❙♦♠❡ ♣r♦♣❡rt✐❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❥♦✐♥✐♥❣s ✐♥ ❡r❣♦❞✐❝ t❤❡♦r②✱ ✐♥ ❊r❣♦❞✐❝ t❤❡♦r② ❛♥❞ ✐ts ❝♦♥♥❡❝✲t✐♦♥s ✇✐t❤ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ✭❆❧❡①❛♥❞r✐❛✱ ✶✾✾✸✮✱ ✈♦❧✳ ✷✵✺ ♦❢ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ▲❡❝t✉r❡ ◆♦t❡ ❙❡r✳✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✶✾✾✺✱ ♣♣✳ ✷✵✼✕✷✸✺✳

❬✹✶❪ ❈✳ ❯❧❝✐❣r❛✐✱ ▼✐①✐♥❣ ♦❢ ❛s②♠♠❡tr✐❝ ❧♦❣❛r✐t❤♠✐❝ s✉s♣❡♥s✐♦♥ ✢♦✇s ♦✈❡r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❊r❣♦❞✐❝❚❤❡♦r② ❉②♥❛♠✳ ❙②st❡♠s✱ ✷✼ ✭✷✵✵✼✮✱ ♣♣✳ ✾✾✶✕✶✵✸✺✳

❬✹✷❪ ✱ ❲❡❛❦ ♠✐①✐♥❣ ❢♦r ❧♦❣❛r✐t❤♠✐❝ ✢♦✇s ♦✈❡r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❏✳ ▼♦❞✳ ❉②♥✳✱ ✸ ✭✷✵✵✾✮✱ ♣♣✳ ✸✺✕✹✾✳❬✹✸❪ ✱ ❆❜s❡♥❝❡ ♦❢ ♠✐①✐♥❣ ✐♥ ❛r❡❛✲♣r❡s❡r✈✐♥❣ ✢♦✇s ♦♥ s✉r❢❛❝❡s✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✭✷✮✱ ✶✼✸ ✭✷✵✶✶✮✱ ♣♣✳ ✶✼✹✸✕✶✼✼✽✳❬✹✹❪ ❲✳ ❆✳ ❱❡❡❝❤✱ ■♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❏♦✉r♥❛❧ ❞✬❆♥❛❧②s❡ ▼❛t❤é♠❛t✐q✉❡✱ ✸✸ ✭✶✾✼✽✮✱ ♣♣✳ ✷✷✷✕✷✼✷✳❬✹✺❪ ✱ ●❛✉ss ♠❡❛s✉r❡s ❢♦r tr❛♥s❢♦r♠❛t✐♦♥s ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ ♠❛♣s✱ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✶✺

✭✶✾✽✷✮✱ ♣♣✳ ✷✵✶✕✷✹✷✳❬✹✻❪ ▼✳ ❱✐❛♥❛✱ ❉②♥❛♠✐❝s ♦❢ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ t❡✐❝❤♠ü❧❧❡r ✢♦✇s✳ ❆✈❛✐❧❛❜❧❡ ❢r♦♠ ❤tt♣✿✴✴✇✸✳✐♠♣❛✳

❜r✴⑦✈✐❛♥❛✳ ▲❡❝t✉r❡ ◆♦t❡s✳❬✹✼❪ ❏✳✲❈✳ ❨♦❝❝♦③✱ ❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ ♠❛♣s✿ ❛♥ ✐♥tr♦❞✉❝t✐♦♥✱ ✐♥ ❋r♦♥t✐❡rs ✐♥ ♥✉♠❜❡r

t❤❡♦r②✱ ♣❤②s✐❝s✱ ❛♥❞ ❣❡♦♠❡tr②✳ ■✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✷✵✵✻✱ ♣♣✳ ✹✵✶✕✹✸✺✳❬✹✽❪ ❆✳ ❩♦r✐❝❤✱ ❋✐♥✐t❡ ●❛✉ss ♠❡❛s✉r❡ ♦♥ t❤❡ s♣❛❝❡ ♦❢ ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ▲②❛♣✉♥♦✈ ❡①♣♦♥❡♥ts✱ ❆♥♥✳ ■♥st✳

❋♦✉r✐❡r✱ ●r❡♥♦❜❧❡✱ ✹✻ ✭✶✾✾✻✮✱ ♣♣✳ ✸✷✺✕✸✼✵✳❬✹✾❪ ❆✳ ❩♦r✐❝❤✱ ❉❡✈✐❛t✐♦♥ ❢♦r ✐♥t❡r✈❛❧ ❡①❝❤❛♥❣❡ tr❛♥s❢♦r♠❛t✐♦♥s✱ ❊r❣♦❞✐❝ ❚❤❡♦r② ❉②♥❛♠✳ ❙②st❡♠s✱ ✶✼ ✭✶✾✾✼✮✱ ♣♣✳ ✶✹✼✼✕

✶✹✾✾✳❬✺✵❪ ✱ ❍♦✇ ❞♦ t❤❡ ❧❡❛✈❡s ♦❢ ❛ ❝❧♦s❡❞ 1✲❢♦r♠ ✇✐♥❞ ❛r♦✉♥❞ ❛ s✉r❢❛❝❡❄✱ ✐♥ Ps❡✉❞♦♣❡r✐♦❞✐❝ t♦♣♦❧♦❣②✱ ✈♦❧✳ ✶✾✼ ♦❢ ❆♠❡r✳

▼❛t❤✳ ❙♦❝✳ ❚r❛♥s❧✳ ❙❡r✳ ✷✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✶✾✾✾✱ ♣♣✳ ✶✸✺✕✶✼✽✳

❆❞❛♠ ❑❛♥✐❣♦✇s❦✐P❡♥♥ ❙t❛t❡ ❯♥✐✈❡rs✐t② ▼❛t❤❡♠❛t✐❝s ❉❡♣❛rt♠❡♥t✱ ❯♥✐✈❡rs✐t② P❛r❦✱❙t❛t❡ ❈♦❧❧❡❣❡✱ P❆ ✶✻✽✵✷✱ ❯❙❆❛❞❦❛♥✐❣♦✇s❦✐❅❣♠❛✐❧✳❝♦♠

❏♦❛♥♥❛ ❑✉➟❛❣❛✲Pr③②♠✉s■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ P♦❧✐s❤ ❆❝❛❞❛♠② ♦❢ ❙❝✐❡♥❝❡s✱ ➅♥✐❛❞❡❝❦✐❝❤ ✽✱ ✵✵✲✾✺✻ ❲❛rs③❛✇❛✱ P♦❧❛♥❞❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ◆✐❝♦❧❛✉s ❈♦♣❡r♥✐❝✉s ❯♥✐✈❡rs✐t②✱ ❈❤♦♣✐♥❛ ✶✷✴✶✽✱ ✽✼✲✶✵✵❚♦r✉➠✱ P♦❧❛♥❞❥♦❛♥♥❛✳❦✉❧❛❣❛❅❣♠❛✐❧✳❝♦♠

❈♦r✐♥♥❛ ❯❧❝✐❣r❛✐❙❝❤♦♦❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❇r✐st♦❧✱ ❍♦✇❛r❞ ❍♦✉s❡✱ ◗✉❡❡♥s ❆✈❡✱ ❇❙✽ ✶❙◆❇r✐st♦❧✱ ❯♥✐t❡❞ ❑✐♥❣❞♦♠❝♦r✐♥♥❛✳✉❧❝✐❣r❛✐❅❜r✐st♦❧✳❛❝✳✉❦