stable ergodicity beyond partial hyperbolicity1 mechanisms that activate stable ergodicity partial...

Stable ergodicity beyond partialhyperbolicity

Jana Rodriguez HertzSUSTech - China

Dynamics on the screenAugust 5, 2020

1 introduction

1 setting

setting

• M compact Riemannian manifold without boundary• m volume probability measure• Diffr

m(M): Cr -diffeomorphisms preserving m

1 stable ergodicity

stable ergodicity

• f ∈ Diff1m(M) is stably ergodic

• if there exists f ∈ U ⊂ Diff1m(M) open

• such that

g ∈ U ∩ Diff2m(M) ⇒ g ergodic

1 question

question

• does there exist U ⊂ Diff1m open

• such that

g ∈ U ⇒ g ergodic?

1 mechanisms that activate stable ergodicity

hyperbolicity

f ∈ Diff1m(M) is Anosov or hyperbolic if

• there exists a Df -invariant splitting TM = Es ⊕ Eu

• such that

TM = Es ⊕ Eu

↓ ↓Df − contracting Df − expanding


Anosov-Sinai (1967)

• f ∈ Diff1+αm (M) hyperbolic

• ⇒ ergodic• ⇒ stably ergodic


Grayson - Pugh - Shub (1995)

• ∃ a non-hyperbolic stably ergodic diffeomorphism


partial hyperbolicity

f ∈ Diff1m(M) is partially hyperbolic if

• there exists a Df -invariant splittingTM = Es ⊕ Ec ⊕ Eu

• such that

TM = Es ⊕ Ec ⊕ Eu

↓ ↓ ↓contracting intermediate expanding


conjecture: pugh - shub (1995)

stable ergodicity is Cr -dense among partially hyperbolicdiffeomorphisms


Avila - Crovisier - Wilkinson (2016)

stable ergodicity is C1-dense among partially hyperbolicdiffeomorphisms

Hertz - H. - Ures (2008)

stable ergodicity is C∞-dense among partially hyperbolicdiffeomorphisms (c = 1)


Tahzibi (2004)

• ∃ a non-partially hyperbolic stably ergodicdiffeomorphism


dominated splitting

f ∈ Diff1m(M) admits a dominated splitting

• there exists a Df -invariant splitting TM = E ⊕ F• such that

TM = E ⊕ F↓ ↓

more contracting more expanding


it is a necessary condition:

stable ergodicity

⇓

dominated splitting


pugh - shub (1995)

a little hyperbolicity goes a long way toward guaranteeingstable ergodicity

1 conjecture

conjecture: JRH. (2012)

generically in Diff1m(M)

dominated splitting

⇓

stable ergodicity

2 exploring new mechanisms for SE

2 known mechanism for PHD

accessibility

• for partially hyperbolic diffeomorphisms• Pugh - Shub proposed accessibility

•

2 known mechanism for PHD

pugh - shub program

• for partially hyperbolic diffeomorphisms in Diff2m(M)

• stable accessibility is Cr -dense• accessibility⇒ ergodicity

2 other mechanisms activating SE

other mechanisms

• in dimension 3• dominated splitting⇒ one hyperbolic bundle• ⇒ either E = Es or F = Eu

• suppose TM = E ⊕ Eu

• there is an invariant foliation Fu tangent to Eu


minimality

• a foliation is minimal• if every leaf is dense


program in dimension 3

• for f ∈ Diff1m(M) with a dominated splitting

• (stable) minimality of the Fu is C1-dense• C1-generically, minimality of Fu ⇒ stable ergodicity


theorem (G. Nuñez, JRH 2019)

generically in Diff1m(M3),

• Fu minimal

⇒ stable ergodicity

2 question

question (2014)

generically in Diff1m(M3),

• dominated splitting• ⇒ Fu or Fs minimal?


question - in any dimension

Fu minimal

⇓

stable ergodicity

2 conjecture

conjecture JRH (2019)

generically in Diff1m(M)

Fu minimal

⇓

stable ergodicity

2 theorem

G. Núñez - JRH (2020)

for a generic map f ∈ Diff1m(M), if

• Fu minimal

⇒ ∃ U(f ) ⊂ Diff1m(M): ∀g ∈ Diff2

m(M) ∩ U

• ∃ ergodic component Phc(q)

• Phc(q)ess

= M

2 another mechanism for SE

G. Núñez - D. Obata - JRH (2020)

generically in Diff1m(M3)

• dominated splitting• + a little partial hyperbolicity (∗)• ⇒ stable ergodicity

2 another mechanism for SE

G. Núñez - D. Obata - JRH (2020)

• in the isotopy class of every partially hyperbolicf ∈ Diff1

m(M3)

• there is a stably ergodic diffeomorphism• which is not (strictly) partially hyperbolic

3 ergodic homoclinic classes

3 Pesin invariant manifolds

Pesin stable/unstable manifolds

W−(x) =

{y ∈ M : lim sup

n→∞

1n

log d(f n(x), f n(y)) < 0}

W+(x) =

{y ∈ M : lim sup

n→∞

1n

log d(f−n(x), f−n(y)) < 0}

3 ergodic homoclinic class

ergodic homoclinic class (HHTU11)

p ∈ PerH(f )

Phc+(p) = {x : W+(x) t W s(o(p)) 6= ∅}

W s(p)

Wu(p) W+(x)

p

3 ergodic homoclinic class

ergodic homoclinic class (HHTU11)

p ∈ PerH(f )

Phc−(p) = {x : W−(x) t W u(o(p)) 6= ∅}

W s(p)

Wu(p) W−(x)

x

p

3 criterion of ergodicity

criterion of ergodicity (HHTU11)

• p ∈ PerH(f ), f ∈ Diff2m(M)

• m(Phc−(p)) > 0 and m(Phc+(p)) > 0

then

1 Phc−(p)◦= Phc+(p) :

◦= Phc(p)

2 f |Phc(p) ergodic

4 strategy

4 strategy

strategy

prove that C1-densely, for f ∈ Diff1m(M)

• ∃ p ∈ PerH(f )

• ∃ U ⊂ Diff1m(M)

• such that g ∈ U ∩ Diff2m(M)⇒

m(Phc(p)) = 1

4 strategy

setting

• f ∈ Diff1m(M3)

• with TM = E ⊕ Eu dominated splitting• ∃ p ∈ PerH(f ) with u(p) = dim(Eu) = 1

4 strategy

strategy

on the one hand, use geometric approach to find• ∃ U ⊂ Diff1

m(M) such that• g ∈ U ∩ Diff2

m(M)⇒

m(Phcu(p)) = 1

• ⇒ m(Phc+(p)) = 1

4 strategy

strategy

on the other hand, use generic approach to guarantee• ∃ U ⊂ Diff1

m(M) such that• g ∈ U ∩ Diff2

m(M)⇒

m(Phc−(p)) > 0

• thenPhc+(p)

◦= Phc−(p)

◦= Phc(p)

◦= M

4 strategy

generic mechanism - M - B - H - ACW - AB

for a generic f ∈ Diff1m(M) with dominated splitting

• f ergodic• ∃q ∈ PerH(f ) such that Phc(q)

◦= M

• Oseledets splitting is dominated u(q) = #{LE > 0}• ∃U(f ) such that m(Phc(q)) > 0 for all g ∈ U

4 strategy

geometric mechanism - minimality

assume f ∈ Diff1m(M3) with dominated splitting TM = E ⊕

Eu generic• ⇒ ∃p ∈ PerH(f ) with u(p) = dim Eu = 1• if Fu is minimal• ∃ U(f ) such that for all g ∈ U

Phcu(p) = M

4 strategy



Eu generic• if u(p) = u(q)

• generically Phc(p) = Phc(q) (HHTU - AC)• on one hand Phc+(p) = M for all g ∈ U• on the other hand Phc−(p) > 0 for all g ∈ U• ⇒ f stably ergodic

4 strategy



Eu generic• if u(q) > u(p)

• TM = Es ⊕ Ec ⊕ Eu

• ⇒ f partially hyperbolic• ⇒ f stably ergodic

4 strategy

geometric mechanism - a little partial hyperbolicity


Eu generic• ∃ p ∈ PerH(f ) with u(p) = dim(Eu) = 1• ⇒ Phcu(p) is open• assume

Λ(f ) = M \ Phcu(p)

is partially hyperbolic (∗)• (NOH hypothesis)

4 strategy


• g 7→ Λ(g) continuous in f• Λ(g) PH• with a generic argument,• we see m(Λ(g)) = 0 for all C2 g ∈ U• ⇒ Phcu(p)

◦= M for all g ∈ U

4 strategy


• u(p) = u(q)

• generically Phc(p) = Phc(q) (HHTU - AC)

• on one hand Phc+(p)◦= M for all g ∈ U

• on the other hand Phc−(p) > 0 for all g ∈ U• ⇒ f stably ergodic

4 strategy

generic argument (if there is time)

• m(Λ(fn)) > 0 with fn → f fn ∈ C2

• ⇒ Λ(fn) su-saturated (P-JRH)• ⇒ Λ(f ) su-saturated• ⇒ ∃ q1,q2 ∈ PerH(f ) such that

W ss(q1) tq W uu(q2) 6= ∅ (H)

• ⇒ non-generic situation (Kupka-Smale argument)

4 question

question

stable ergodicity

⇓?

mixing

thank you!

thank you!

stable ergodicity beyond partial hyperbolicity1 mechanisms that activate stable ergodicity partial...

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