x µŒ ˘˘ µ Œ 6 µ œ fiµÙ fˇµ2013 c 4maths.nju.edu.cn/~yangfei/papers/ph.d thesis.pdf ·...
TRANSCRIPT
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1Ù XØ 1
1.1 Vã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ̽n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Julia 8 Cantor ±kn¼ê . . . . . . . . . . . . . . . . . . 2
1.2.2 McMullen N Julia 8[é¡x . . . . . . . . . . . . . . . . . 5
1.2.3 x¼êÄåXÚ . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 ng tableaux ¢y½n . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.5 McMullen NááÚ>. Hausdorff ê . . . . . . 9
1.3 ÎÒ(² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1Ù E©ÛÚEÄåXÚ£ý 11
2.1 EA, [/N, ÿ Riemann N½n . . . . . . . . . . . . . . . . 11
2.2 Riemann N½n, Koebe ½n, . . . . . . . . . . . . . . . 13
2.3 ±Ï:, Fatou 8, Julia 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Ak^Ún . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1nÙ Julia 8 Cantor ±kn¼ê 18
3.1 Úó . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 .: ±9V/ . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Cantor ±. Julia 8mÿÀÝ . . . . . . . . . . . . . . . . . . . . 30
3.4 Julia 8þÿÀd Cantor ±kn¼êÝa . . . . . . . . . . . . 34
3.5 Cantor ±kn¼ê Julia 8þÿÀÝaêO . . . . . . . . . 36
3.6 Julia 8 Cantor ±Vkn¼ê . . . . . . . . . . . . . . . . . . 40
3.7 õV Cantor ±. Julia 8~f . . . . . . . . . . . . . . . . . 46
3.8 Ún 3.7.2 y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1oÙ McMullen N Julia 8[é¡x 56
4.1 cóÚĽ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 gd.:<º/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
i
4.2.1 <ºn©½nÚëêm©) . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Cantor 8[é¡üz . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.3 Cantor ±[é¡üz . . . . . . . . . . . . . . . . . . . . . 60
4.2.4 Sierpinski /#[é¡üz . . . . . . . . . . . . . . . . . . . . 63
4.3 gd.:<º/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 nØ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.2 ÓN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.3 <º¹e Sierpinski /#[é¡üz . . . . . . . . . . . . 66
4.4 Cantor ±[é¡AÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
1ÊÙ x¼êÄåXÚ 71
5.1 cóÚy²g´Vã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Ä(J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 5z« . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 äk û[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5 ÃâNEÚ§ëY5 . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 ½n 5.1.1 y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
18Ù ng tableaux ¢y½n#y² 96
6.1 Ä tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2 äþÄåXÚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 ng children Nòÿ . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 lIP:Ðng children N . . . . . . . . . . . . . . . . . . . . 101
6.5 IP:ÚÐng children Nê8 . . . . . . . . . . . . . . . . . 104
1ÔÙ McMullen NááÚ>. Hausdorff ê 105
7.1 ½nã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 úªí . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3 ON¹ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
ë©z 114
®uLÚ®¤Ø© 119
120
Julia 8ÿÀÚAÛ5ïÄ
¥©Á
X¼ê Julia8ÿÀÚAÛ5ïÄ´EÄåXÚ¥¯K.ùÙ
¥Ì) Julia8 (½öÙf8)ëÏ5,ÛÜëÏ5±9K5SN.éuõª
Julia 8ÿÀ5, éõöó. Äk´gõª fc(z) = z2 + c, Ù¥
c áu Mandelbrot 8, Yoccoz y²XJ fc شá, K fc Julia 8 J(fc)
´ÛÜëÏ [38]. Lyubichy² c¢ê, J(fc)ÛÜëÏ [46]. ëê c¦ fc
äk Siegel, Peterseny²XJ fc 3 Siegelþ^=êk..,K J(fc)
´ÛÜëÏ [57]. XJ^=ê David a., Petersen Ú Zakeri y² Siegel
>.´ Jordan éA Julia 8´ÛÜëÏ [58]. d, Avila, Buff Ú Cheritat
y²3>.1w Siegel ~f [4].
éungõª, Branner Ú Hubbard y²ØêõëÏ©|, Ù Julia 8
ëÏ©|Ñ´ü: [11]. C, ©z [63] òù(Øí2?¿õª. ékn¼ê
5`, XJ§´AÛk, @oØêõëÏ©|±, Ù Julia 8ëÏ©|o
´ü:, o´^ Jordan [59].
1¦ Julia 8ØëÏ¿Ù¤këÏ©|Ñ´ Jordan kn¼ê~
f´ McMullen é [49]. ¦y²XJ f(z) = z2 + λ/z3 ¥ λ v, K f
Julia8ÓuIOn© Cantor8ü ±¦È.ù«a. Julia8y3¡
Cantor±. , õöļêx fλ(z) = zm +λ/zl, Ù¥ l,m ≥ 2
λ ∈ C \ 0, y3Ñ¡ McMullen N. N´`² 1/l + 1/m < 1 λ é,
fλ Julia 8´ Cantor ± ([49, §7], [24, §3]).
ùØ©1Ü©ÌïÄ Julia 8 Cantor ±kn¼ê. 8c=3
·ëê¦ McMullen N (½3Ù Fatou 8þ6Äkn¼ê) Julia 8´
Cantor ±. ùr¦·g´Ä3Ù§kn¼ê¦Ù Julia 8 Cantor
±. , HaıssinskyÚ PilgrimÏL[/ÃâEÑa “þ” McMullenN
ØÓ Julia 8 Cantor ±kn¼ê, ¦¿vkÑùkn¼êLª
[35]. Ï g,¯KÒ´UÄÑùkn¼êLª.
¯¢þ, 3ùØ©¥, ·ØÑùa.kn¼êäNLª,
3 “þ” é¤k Julia 8 Cantor ±kn¼ê. ùp¤¢ “þ” ´
3Ä Julia þÿÀÝa¿Âe. äN/`, ·éxkn¼êL
ª (McMullen N´ùx¼êAÏ/), ¦·ëê, § Julia 8Ñ´
Cantor ±. ,¡, éu?¿½ Julia 8 Cantor ±kn¼ê, ·o
±3éùxkn¼ê¥é¼ê¯k½¼ê¦§3éA Julia
8þ´ÿÀÝ (½n 3.1.1 Ú 3.1.2).
3dóÄ:þ, ·é Julia 8 Cantor ±kn¼ê3 Julia 8þÿ
ÀÝaê8ÑOúªÚþe.O, ¿é 5 ≤ d ≤ 36 /ÑäNê
L, Ù¥ d kn¼êNÝ.
Vkn¼êäküÄåXÚ5, ·éXVkn¼ê¦
§ Julia 8´ Cantor ±. â·¤, ù´1 Julia 8 Cantor ±V
kn¼ê~f. ÙÌEg´ò5Vkn¼êáÚ^üëÏÔ
O.
Ø©1Ü©ÌïÄ McMullen N Julia 8AÛ5. â<ºn©½
n, XJ McMullen Ngd.:Ñá¤áÚ, @o§ Julia 8U´
Cantor 8, Cantor ±½ö´ Sierpinski /# [24]. ·y²3ù«¹
e, McMullen N Julia 8o[é¡duIOn© Cantor 8, IO
Cantor ±½ö´ Sierpinski /# (3,«¿Âe´IO).
éu McMullen N fλ ¥ëê λ áu¢ê/, ©z [62, 76] ¥Ñ fλ
Julia8 Jλ ´ Sierpinski/#¿^.3dÄ:þ,ö3©¥Ñ Jλ [é
¡du Sierpinski /#¿^. AO/, 3Vkn¼ê, ¦Ù Julia
8[é¡du Sierpinski/#. d, éuëê λáuEê/, ·Ñ Jλ
[é¡du Sierpinski /#¿©^ (d¦ fλ Lª¥üê
÷v l = m ≥ 3).
lÿÀÝ5w, ¤k Cantor ±Ñ´, ù´Ï§ÑÿÀd (Ó
) u “IO” Cantor ± C × S1, Ù¥ C n© Cantor 8 S1 ü ±. Ï
d, ¤k Cantor ±¤8Ü´L(, ·±^[é¡AÛÝ
5w. ¯¢þ, [é¡AÛp¡Ä¯KÒ´äü½ÓÝþm´
Ä´*d[é¡d. Ýþm/ê½Â¤k[é¡dÝþ
m Hausdorff êe(.. âéØ©1Ü©éÑ Julia 8 Cantor ±
kn¼êx|Ü©Û, ·`²3ùx¼ê¥3~f, ¦§ Julia 8
McMullen Julia 8´ÓØ´[é¡d. AO/, ù~f±^5äN/
y3 Julia 8 Cantor ±Vkn¼ê¦Ù/ê?¿Cu 2.
Ø©1nÜ©ïÄx¼ê Siegel >.K5. X¼ê Siegel
>. (´ Julia 8f8) ÿÀÚAÛ5´~¯K. é^=ê
\þ½â^, Douady ß Siegel >.½´ Jordan . 3ù
¯Kþ, Douady, Zakeri, Shishikura Ú Zhang éukn¼ê/Ñz (
[26, 77, 66, 80]). éu¼ê/ù¯Kvk)û ( [32, 41, 78, 79]).
öļêx fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0, ¿y² θ k
.., ¼ê fα ±:% Siegel >.o´[±²L 2 , 4 , ½ 6
.:.
Ø©1oÜ©Ä. tableau ¢y¯K. .IP tableau ´ Branner Ú
Hubbard 3ïÄngõªÚ\ [11]. ù´~k^óä, §3ïÄ Julia 8
ÿÀ5¥åX©^. Branner Ú Hubbard y²: XJÄ.I
P tableau ÷vK, @où tableau ½±dngõª¢y. ·^d
õªpûäþÄåXÚÚäÄåXÚ¢y½n ( [30, 18]) Ñ Branner Ú
Hubbard tableau ¢y½n#y².
d, 3Ø©Ü©öïÄ McMullen NááÚ>
.5, ¿ÑÙ Hausdorff êìCLª.
'c: Julia 8, Fatou 8, Cantor ±, Sierpinski /#, McMullen N, [é¡d,
Hausdorff ê, Siegel , aõªä, tableau
2000 MR ÌK©a: 37F45, 37F20, 37F10, 37F25, 32G05
¥ã©aÒ: O174.5
The Study of the Topological and GeometricProperties of Julia Sets
ABSTRACT
The study of the topological and geometric properties of the Julia sets of holomorphic
maps is an important problem in complex dynamics. It mainly includes the connected-
ness, locally connectivity and the regularity of Julia sets (or their subsets), etc. For the
topological properties of the Julia sets of polynomials, many authors contributed on this.
For the quadratic polynomials fc(z) = z2 + c, where c belongs to the Mandelbrot set,
Yoccoz proved that if fc is not infinitely renormalizable, then the Julia set J(fc) of fc is
locally connected [38]. Lyubich proved that if c is real, then J(fc) is locally connected
[46]. For the case when fc has a Siegel disk, Petersen proved that if the rotation number
on the Siegel disk is of bounded type, then J(fc) is also locally connected [57]. If the
rotation number is of David type, then Petersen and Zakeri proved that the boundary of
the Siegel disk is a Jordan curve and the Julia set is locally connected [58]. Moreover,
Avila, Buff and Cheritat proved that there exists Siegel disk whose boundary is a smooth
curve [4].
For cubic polynomials, Branner and Hubbard proved that all but countably many
components of the Julia set are single points. Recently, this result has been extended to
all polynomials in [63]. For the rational maps, if it is geometrically finite, then, with the
possible exception of countable components, every Julia component is either a point or a
Jordan curve [59].
The first example of rational map with disconnected Julia set whose components are
all Jordan curves was discovered by McMullen [49]. He showed that if f(z) = z2 + λ/z3
and λ is small enough, then the Julia set of f is homeomorphic to the product of the
middle third Cantor set and the unit circle. These types of Julia sets are called Cantor
circles. Later, many authors focus on the family, which is commonly referred as the
McMullen maps: fλ(z) = zm + λ/zl, where l,m ≥ 2 and λ ∈ C \ 0. It can be shown
that when 1/l + 1/m < 1 and λ is small enough, the Julia set of fλ is a Cantor set of
circles (see [49, §7] and [24, §3]).
The first part of this thesis mainly focuses on the rational maps whose Julia sets
are Cantor circles. Up to now, it is known that McMullen maps (or the rational maps
after perturbing on their Fatou sets) are the only examples such the Julia sets of rational
maps are Cantor circles. This motivates us to think about the question whether there
exist other rational maps whose Julia sets are Cantor circles. Although Haıssinsky and
Pilgrim have constructed a class of rational maps whose Julia sets are Cantor circle which
is different from McMullen maps “essentially” by quasiconformal surgery. However, they
did not give the specific expressions of these maps [35]. Therefore, a natural question is
whether one can give the specific expressions of these maps.
In fact, in this thesis, we not only give the specific expressions of these types of
rational maps, but also find out all the rational maps whose Julia sets are Cantor circles
“essentially”. The word “essentially” means we consider this problem in the sense of the
topological conjugacy classes on the Julia sets. Specifically, we find the specific expressions
of a family of rational maps (McMullen maps are the special cases) such that their Julia
sets are Cantor circles after choosing appropriate parameters. On the other hand, for
each given rational map whose Julia set is a Cantor set of circles, we can always find a
map in the family which we have found such they are topologically conjugate on their
corresponding Julia sets (Theorems 3.1.1 and 3.1.2).
Based on this, we calculate and give a lower and upper bound of the number of
different topological conjugacy classes on the Julia sets of rational maps whose Julia sets
are Cantor circles. Moreover, we give the specific numbers of the different topological
conjugacy classes for 5 ≤ d ≤ 36, where d is the degree of the rational maps.
Hyperbolic rational maps possess simple dynamical properties. We find out a series
of non-hyperbolic rational maps whose Julia sets are Cantor circles. As far as we known,
this is the first example of non-hyperbolic rational maps whose Julia sets are Cantor
circles. The idea behind the construction is the attracting basin of the original rational
maps has been replaced by a simply connected parabolic basin.
The second part of this thesis mainly focuses on the study of the geometric properties
of the Julia sets of McMullen maps. According to the Escape Trichotomy Theorem, if all
the critical points of a McMullen map are attracted by ∞, then the corresponding Julia
set is either a Cantor set, a Cantor set of circles or a Sierpinski carpet [24]. We prove
that in this case, the Julia set of a McMullen is quasisymmetrically equivalent either to a
standard Cantor set, a standard Cantor set of circles or a round Sierpinski carpet (which
is also standard in some sense).
For the case when the parameter of the McMulle family is real, a sufficient and
necessary condition such the Julia set Jλ of McMullen map fλ is a Sierpinski carpet was
gave in [62, 76]. Based on this, we give a sufficient and necessary condition to guarantee
that Jλ is quasisymmetrically equivalent to a round Sierpinski carpet. In particular, there
exists non-hyperbolic rational map whose Julia set is quasisymmetrically equivalent to
a round Sierpinski carpet. Moreover, for case when λ is complex, we give a sufficient
condition to guarantee that Jλ is quasisymmetrically equivalent to a round Sierpinski
carpet (but we need to set l = m ≥ 3, where l,m is the integers in fλ).
From the topological point of view, all Cantor sets of circles are the same since they
are all topologically equivalent (homeomorphic) to the “stand” Cantor set of circles C×S1,
where C is the middle third Cantor set and S1 is the unit circle. Therefore, to obtain much
richer structure of all Cantor sets of circles, we can look at the Cantor circles equipped
with metric from the point of view of quasisymmetric geometry. In fact, a basic problem
in the quasisymmetric geometry is to determine whether two given homeomorphic metric
spaces are quasisymmetrically equivalent to each other. The conformal dimension of a
metric space is the infimum of the Hausdorff dimensions of all metric spaces which are
quasisymmetrically equivalent to the original metric space. According to the combina-
torial analysis on the rational maps which we have found in the first part of this thesis,
we show that this family gives a series of specific examples, such the Julia sets of them
are homeomorphic, but not quasisymmetrically equivalent to the Julia sets of McMullen
maps. In particular, these examples can be used to verify that there exist hyperbolic
rational maps whose Julia sets are Cantor circles and whose conformal dimensions are
arbitrarily close to 2.
The third part of this thesis studies the regularity of the boundaries of a family of
entire functions. The study of the topological and geometric properties of the boundaries
of Siegel disks (subsets of Julia sets) is an important problem. After assuming the rotation
number satisfies some arithmetical condition, Douady conjectured that the boundary of
the Siegel disk must be a Jordan curve. For this problem, Douady, Zaker, Shishikura and
Zhang contributed to the rational case (see [26, 77, 66, 80]). But this problem is far from
being solved for transcendental case (see [32, 41, 78, 79]). We consider the one-dimensional
family fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0, and prove that the boundary of the
Siegel disk centered at the origin of fα is a quasicircle which passes through 2, 4 or 6
critical points of fα counted with multiplicity if θ is of bounded type.
The fourth part of this thesis considers the realize problem of the critical tableau. As
a powerful tool, the critical tableau was introduce by Branner and Hubbard when they
studied the cubic polynomials [11]. It plays a central role in the study of locally connec-
tivity of the Julia sets. Branner and Hubbard proved that: If an abstract critical marked
tableau satisfies some rules, then this tableau must be realized by a cubic polynomial. We
give a new proof of the realization theorem of tableaux by using tree dynamics induced
by polynomials and the realization theorem of tree dynamics (see [30, 18]).
In the last part of this thesis, we study the boundary behavior of the immediate basin
of infinity of McMullen maps, and give an asymptotic formula of its Hausdorff dimension.
Keywords: Julia set, Fatou set, Cantor circles, Sierpinski carpet, McMullen map, qua-
sisymmetrically equivalent, Hausdorff dimension, Siegel disk, polynomial-like tree, tableau
2000 MR Subject Classification: 37F45, 37F20, 37F10, 37F25, 32G05
Chinese Library Classification: O174.5
1Ù XØ
§1.1 Vã
ùƬةÌde¡ASN|¤:
1. Julia 8 Cantor ±kn¼êÄåXÚ.
éuùÜ©,·éÑxkn¼ê fp,d1,··· ,dn ,¦Ù¥z3À·
ëê Julia 8Ñ´ Cantor ±, Ù¥ p ∈ 0, 1, n ≥ 2 d1, · · · , dn n ¦∑ni=1
1di< 1 ê. ·òy²XJ n ≥ 3, Kz fp,d1,··· ,dn 3 Julia 8þØ´ÿ
ÀÝu McMullen N. ùL²·éakn¼êäNLª¦§
Julia 8´ Cantor ±, § McMullen N3þ´ØÓ.
éu?¿½ Julia 8 Cantor ±kn¼ê, ·o±3éùxk
n¼ê¥é fp,d1,··· ,dn ¯k½¼ê3éA Julia 8þ´ÿÀÝ. ùL
²·3þé “¤k” Julia 8 Cantor ±kn¼ê. dd, ·é Julia
8 Cantor ±kn¼ê3 Julia 8þÿÀÝaê8ÑOúªÚ
þe.O, ¿é 5 ≤ d ≤ 36 /ÑäNêL, Ù¥ d kn¼êNÝ.
Vkn¼êÄåXÚ5'ü, ·éXVkn¼ê¦
§ Julia 8´ Cantor ±. â·¤, ù´1 Julia 8 Cantor ±
Vkn¼ê~f. ÙÌEg´ò5Vkn¼êáÚ^üëÏ
Ô5O.
2. McMullen N Julia 8[é¡x.
ùÜ©Ì) McMullenN fλ(z) = zm +λ/zl Julia8[é¡AÛx,
Ù¥ l,m ≥ 2 λ ∈ C \ 0 . ·y²XJ fλ gd.:Ñá¤áÚ, @o
§ Julia 8o[é¡duIOn© Cantor 8, IO Cantor ±½
ö´IO Sierpinski /#.
éu fλ ¥ëê λ áu¢ê/, ·3©¥Ñ Jλ [é¡duIO
Sierpinski /#¿^. éuëê λ Eê l = m ≥ 3 /, ·Ñ Jλ [
é¡du Sierpinski /#¿©^.
du?¿ Cantor ±ÑÓuIOn© Cantor 8ü ±¦È. ·
Ä Cantor ±[é¡[é¡d¯K. âØ©1Ü©é¼êx, ·Ñ
þVkn¼ê Julia ÓØ[é¡d~f. d, âé·éÑ Julia 8
Cantor±kn¼êx|Ü©Û,·`²ù¼êxTÐÑäN~
f5y3 Julia 8 Cantor ±Vkn¼ê¦Ù/ê?¿Cu 2.
3. x¼êÄåXÚ.
1
2 1Ù XØ
3ùÜ©¥, ·ïÄx¼ê Siegel >.K5. ·Ä¼ê
x fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0, ¿y² θ k.., ¼ê fα ±
:% Siegel >.o´[±²L 2 , 4 , ½ 6 .:.
4. ng tableaux ¢y#y².
3ùÜ©¥, ·Ä. tableau ¢y¯K. Branner Ú Hubbard y²: XJ
Ä.IP tableau÷v,K,@où tableau½±dngõ
ª¢y. ·^dõªpûäþÄåXÚÚäÄåXÚ¢y½nÑ Branner
Ú Hubbard tableau ¢y½n#y².
5. McMullen NááÚ>. Hausdorff êìCúª.
Ø©Ü©,·Ñëê λªu 0, McMullenN fλ(z) = zn+λ/zn
ááÚ>. Hausdorff êìCúª, Ù¥ n ≥ 3.
§1.2 ̽n
§1.2.1 Julia 8 Cantor ±kn¼ê
XJ Riemann ¥¡ C f8Óu C × S1, Ù¥ C Cantor n©8 S1
ü ±, KTf8¡´ Cantor ±. éu McMullen N:
fλ(z) = zm + λ/zl, (1.2.1)
®² 1/l + 1/m < 1 λ é, ¼ê fλ Julia 8´ Cantor ± (l =
3,m = 2 ¹ [49, §7] ¹ [24, §3]).
y3kXeng,¯K:
(1) Ø McMullen N, ´Ä3Ù§kn¼êÙ Julia 8 Cantor ±?
(2) XJ1¯KY´½, @§´o? ó, ·UÄéѧ
Lª?
(3) ·UÄ3½§ÝþéѤk Julia 8 Cantor ±kn¼ê?
e¡½n 1.2.1 Ú 1.2.2 Ñþ¡ù¯K½£. ·ò31nÙÑù
¯K[?Ø. - p ∈ 0, 1, n ≥ 2, d1, · · · , dn n ¦∑n
i=11di< 1
ê. ½Â
fp,d1,··· ,dn(z) = z(−1)n−pd1
n−1∏i=1
(zdi+di+1 − adi+di+1
i )(−1)n−i−p , (1.2.2)
Ù¥ a1, · · · , an−1 n− 1 ÷v 0 < |a1| < · · · < |an−1| < 1 Eê.
§1.2. ̽n 3
½n 1.2.1. é? p ∈ 0, 1, n ≥ 2, ±9¦∑n
i=11di< 1 n ê
d1, · · · , dn, Ñ3·ëê ai, ¦ fp,d1,··· ,dn Julia 8´ Cantor ±, Ù¥ 1 ≤i ≤ n− 1.
3 §3.2 ¥, ·òÑëê ai äN. d, ·òy²XJ n ≥ 3, K?
Û fp,d1,··· ,dn ÚMcMullenN3§éA Julia8þÑØ´ÿÀÝ (½n 3.2.6).
ùL²·éakn¼êäNLª¦§ Julia 8´ Cantor ±, §
McMullen N3þ´ØÓ.
~X, - p = 1, n = 4, d1 = d2 = d3 = d4 = 5 ½Â
f1,5,5,5,5(z) =(z10 − a10
1 )(z10 − a103 )
z5(z10 − a102 )
, (1.2.3)
Ù¥ a1 = 0.00025, a2 = 0.005 a3 = 0.1. ÏLO½ö|^½n 3.1.1,
±`² f1,5,5,5,5 Julia 8´ Cantor ±. Ï f1,5,5,5,5 3 Julia 8þÄåXÚ
Ýu 4ÎÒm Σn := 0, 1, 2, 3N ü>²£, fλ 3 Julia8þÄåXÚÝu
2 ÎÒm Σ2 := 0, 1N ü>²£. ùL² f1,5,5,5,5 Ú fλ 3§éA Julia þØ
´ÿÀÝ.
ã 1.1 ¼ê f1,5,5,5,5 Julia 8 (ã), fλ(z) = z3 + 0.001/z3 Julia 8
(mã) Ø´ÿÀÝ. ü Julia 8Ñ´ Cantor ±.
½n 1.2.2. f kn¼ê, Ù Julia 8 Cantor ±. K3 p ∈ 0, 1, ê n ≥ 2 ±9÷v
∑ni=1
1di< 1 d1, · · · , dn Ú·ëê ai ¦ f Ú fp,d1,··· ,dn 3
§éA Julia 8þ´ÿÀÝ, Ù¥ 1 ≤ i ≤ n− 1.
4 1Ù XØ
d½n 1.2.2, ·±é Julia 8 Cantor ±kn¼ê3 Julia 8þÿÀ
Ýaê8ÑOúª.
½n 1.2.3. NÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓÿÀ
Ýaê
N(d) =∑n≥2
](d1, · · · , dn) :n∑i=1
di = d n∑i=1
1
di< 1
+∑n≥3
](d1, · · · , dn) :n∑i=1
di = d,
n∑i=1
1
di< 1, (d1, · · · , dn) = (dn, · · · , d1) n Ûê .
·3L 3.1 ¥Ñ 5 ≤ d ≤ 36 N(d) äNê. d, ·kXeO.
½n 1.2.4. NÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓÿÀ
Ýaê N(d) ÷v
n0∑n=1
(d− n2 − 3n− 1)n
n!≤ N(d) < 2
n1∑n=1
C nd−n−2, (1.2.4)
Ù¥ n0 = [(√
4d+ 5− 3)/2] n1 = [√d− 1]− 1.
XJkn¼ê.:Ñá5±Ï;¤áÚ, KTkn¼ê¡´V.
éu fp,d1,··· ,dn Julia 8©|K5, ±y²XJ fp,d1,··· ,dn ´V, K fp,d1,··· ,dn
z Julia 8ëÏ©|Ñ´[ (íØ 3.3.3).
XJ λ v, K fλ ´V ( [24]). ·±EVkn¼ê¦
§ Julia 8´ Cantor ±. - m,n ≥ 2 ü÷v 1/m + 1/n < 1 ê
λ ∈ C \ 0, ·½Â
Pλ(z) =1n((1 + z)n − 1) + λm+nzm+n
1− λm+nzm+n. (1.2.5)
y 0 ´ Pλ ¦f 1 ÔØÄ:. ·k
½n 1.2.5. XJ 0 < |λ| ≤ 1/(210mn3), K Pλ ´VÙ Julia 8 Cantor
±.
?Ú,·±EõVkn¼ê¦§ Julia8´ Cantor±.
éz n ≥ 2 ±9 1 ≤ i ≤ n, - di = n+ 1. ·½Â
Pn(z) = An(n+ 1)z(−1)n+1(n+1)
nzn+1 + 1
n−1∏i=1
(z2n+2 − b2n+2i )(−1)i−1
+Bn, (1.2.6)
§1.2. ̽n 5
Ù¥ b1, · · · , bn−1 ÷v 1 > |b1| > · · · > |bn−1| > 0 n− 1 Eê
An =1
1 + (2n+ 2)Cn
n−1∏i=1
(1−b2n+2i )(−1)i , Bn =
(2n+ 2)Cn1 + (2n+ 2)Cn
Cn =n−1∑i=1
(−1)i−1b2n+2i
1− b2n+2i
.
ùp An Ú Bn y Pn(1) = 1 P ′n(1) = 1. =, 1 Pn ¦f 1 Ô
ØÄ: (Ún 3.7.1).
½n 1.2.6. éz n ≥ 2 Ú 1 ≤ i ≤ n− 1, XJ |bi| = si, Ù¥ 0 < s ≤ 1/(25n2),
K Pn ´VÙ Julia 8 Cantor ±.
§1.2.2 McMullen N Julia 8[é¡x
duMcMullenN Julia8ÿÀ5®²þ/ïÄ,·éMcMullen
N Julia 8?1AÛþx.
XJ®²üÝþm´Ó, ·'%ùüm´Ä´*d[é¡d
. Ýþm (X, dX) Ú (Y, dY ) ¡´*d[é¡d´3üÓ f : X → Y
Ú η : [0,∞)→ [0,∞) ¦éuzØÓ: x, y, z ∈ X k dY (f(x),f(y))dY (f(x),f(z))
≤ η(dX(x,y)dX(x,z)
).
- l,m ≥ 2 ü÷v 1/l + 1/m < 1 ê, IO Cantor 8 Cl,m ½ÂS
¼êXÚ g0, g1 áÚf, Ù¥ g0(x) = (1− x)/l g1(x) = 1 + (x− 1)/m þ½Â
3ü «m [0, 1]þ. AO/, C3,3 IOn© Cantor8. IO Cantor± Al,m ½
 Cl,m × S1, Ù¥ S1 := z ∈ C : |z| = 1 ´ü ±.
éu McMullen N fλ Julia 8 Jλ ÿÀ5, Devaney, Look Ú Uminsky y
²<ºn©½n. ù½n`XJ fλ ¤k.:Ñ ∞ áÚ, K Jλ o´
Cantor 8, Cantor ±½ö´ Sierpinski /# ( [24, ½n 0.1] ½ö½n
4.2.1). éu Jλ [é¡AÛ5, ·ke¡½n.
½n 1.2.7. b fλ ¤k.:Ñ ∞ áÚ. Ke¡nö¤á:
(1) Jλ ´ Cantor 8, K Jλ [é¡duIOn© Cantor 8 C3,3;
(2) Jλ ´ Cantor ±, K Jλ [é¡duIO Cantor ± Al,m, Ù¥ l
Ú m (1.2.1) ¥ê;
(3) Jλ ´ëÏ, K Jλ [é¡du Sierpinski /#.
ùp Sierpinski/#´ Sierpinski/#z©|>.Ñ´î¼ (½ö
¥¡) ±.
Ø 0Ú∞, N fλ e l+m.:¡´gd.:. ùgd.:
þkÓ; ( §4.2). §oÓáÚ ∞ o;Ók.. XJgd
.:vk ∞ áÚ, K Jλ ´ëÏ ( [25]). 3ù«¹e, üg,¯KÑy:
6 1Ù XØ
(1) Jλ oÿ´ Sierpinski /#?
(2) §´Ä[é¡du/#?
éþ¡¯K, ©z [62] ¥é λ > 0 /Ä. P Kλ = C \ A(∞) fλ
“W¿” Julia 8, Ù¥ A(∞) ∞ áÚ. 31oÙ¥, ·òy²
½n 1.2.8. 3ü¢ê λ0 < λ1 ¦XJ λ ∈ [λ0, λ1], K Jλ [é¡du
/#=e¡÷v:
(I) Kλ SÜ Int(Kλ) = ∅ λ 6= λ0;
(II) λ ∈ H, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿ HØ´ H♥ 3ÓNe.
AO/, XJ λ ∈ (λ0, λ1) ¦ fλ Ù¥.:´îý±Ï, K Jλ [é
¡du/#.
ÓNÚӽ §4.3.2. á=íØ, ·k
íØ 1.2.9. 3Vkn¼ê, Ù Julia 8[é¡du/#.
¯¢þ, â½n 1.2.8 y², ·$3ákn¼ê, Ù Julia
8[é¡du Sierpinski /#.
·òÄ/¦ Jλ ´ Sierpinski /#. AO/, ëê λ ±3E
²¡þCz Ø==´3¢¶þ. , , ù¦·b l = m ≥ 3.
½n 1.2.10. b l = m ≥ 3 fλ gd.;´k.. XJ λ ÷ve¡
^, K Jλ ´ Sierpinski /#.
(a) SÜ Int(Kλ) = ∅ gd.:;4 ∞ áÚ>.Ø;
(b) λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿
H Ø´ H♥ 3ÓNe.
d, XJ λ ÷v (a) ½ö (b) λ 6= rH, K Jλ [é¡du/#.
éz n ≥ 2, - Jp,d1,··· ,dn §1.2.1 ¥½Â fp,d1,··· ,dn Julia 8. 3e5ã
½n¥, ·o´b½ ai ®²ÀЦ Jp,d1,··· ,dn ´ Cantor ±, Ï·=
éù«¹a,. Ó, ·b½ λ v¦ (1.2.1) ¥½Â McMullen N fλ
Julia 8 Jl,m Cantor ±, Ù¥ 1/l + 1/m < 1. XJéz 1 ≤ i ≤ n k
di = n+ 1, ·^ Jn 5L« fp,n+1,··· ,n+1 Julia 8. Ýþm X /êu
¤k X [é¡dÝþm Hausdorff êe(..
§1.2. ̽n 7
½n 1.2.11. Jp,d1,··· ,dn /êu 1 + αd1,··· ,dn, Ù¥ αd1,··· ,dn e¡§
:n∑i=1
d−αd1,··· ,dni = 1.
AO/, XJéz 1 ≤ i ≤ nk di = n+1, K αn := αd1,··· ,dn = log(n)/ log(n+1). XJ
k 6= n, K αk 6= αn. XJ n ≥ 3, Kéz¦ 1/l + 1/m < 1 l,m ≥ 2 k αn 6= αl,m.
â½n 1.2.1 y², ·¤k fp,d1,··· ,dn Ñ´V. 5¿/ê´
Ýþm[é¡ØCþ ( [47]) ?¿Vkn¼ê Julia 8 Hausdorff
êîu 2 ( [70, ½n 4 ÚíØ]). Ïd, ½n 1.2.11 á=ke¡üíØ.
íØ 1.2.12. é?¿ k, n ≥ 2, Julia 8 Jk Ú Jn *d[é¡d= k = n.
d, XJ n ≥ 3, K Jn Ø[é¡du?Û Jl,m, Ù¥ 1/l + 1/m < 1.
íØ 1.2.13. Jn Hausdorff ê Hdim(Jn) ÷v
1 + log(n)/ log(n+ 1) ≤ Hdim(Jn) < 2.
§1.2.3 x¼êÄåXÚ
- f 3:äkÃn¥5ØÄ:5X¼ê. =, f(0) = 0
f ′(0) = e2πiθ, Ù¥ 0 < θ < 1 ´Ãnê. XJ3X¼êò f Ýf5^
= Rθ : z 7→ e2πiθz, K¡ f 3ØÄ: 0 ?´ÛÜ5z. ¼ê f Ýu Rθ
«´üëÏ« ∆f , ¡´ f ±:% Siegel . Ï f 3 Siegel
∆f p¡ÄåXÚ´ü. éõÿ, ·'%>. ∂∆f AÛÚÿÀ5. ~X,
§´Ä´^ Jordan ? ½ö?Ú, §´Ä´[?
XJ^=ê θ ´k.., Douady, Zakeri Ú Shishikura, ©O/y²gõª,
ngõª±9¤kNÝØu 2 õªk.. Siegel >.´[ [26,
77, 66]. éõöé¼ê/Äù¯K. Geyer, Keen Ú Zhang ©O
Ä z 7→ e2πiθzez Ú z 7→ (e2πiθz + αz2)ez [32, 41]. Ó, Cheritat Ä“
ü”¼ê [14].
C, Zakeri Ú Zhang ©O(J. éu5¼ê f(z) =
P (z) exp(Q(z)), Ù¥ f(0) = 0 f ′(0) = e2πiθ, P Ú Q Ñ´õª, Zakeri y²
XJ 0 < θ < 1 ´k..Ãnê, @o f ±:% Siegel >.´
[¹ f .: [78]. 3 [80] ¥, Zhang y²ù(Øé¤kNÝ
Ø$u 2 kn¼ê¤á.
8 1Ù XØ
31ÊÙ¥, ·ò?رϼêx. - 0 < θ < 1 k..Ãn
ê. P
F = fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0. (1.2.7)
éz fα ∈ F , ·y²e¡
½n 1.2.14. ¼ê fα ±:% Siegel >.´[, 3Pê
¿Âe²L fα 2 , 4 ½ö 6 .:.
½ÂBf
Λ = c : |Re(c)| ≤ π/2 \ 0. (1.2.8)
e¡½n 1.2.15 Ñaqu Zakeri ng Siegel õª [77] ±9 Keen-Zhang
¼êx [41] ëêmaqy© (ã 1.2).
ã 1.2 α–²¡ÛÜã, Ù¥ θ = (√
5− 1)/2 ^=ê. ¥mçÚÜ©L«
Γint, ÚÜ©K¹3 Γext ¥. >. Γ ´^ßü4. ã¡
: [−0.20, 0.35]× [−0.25, 0.30].
½n 1.2.15. - F = fα : α ∈ C \ 0 (1.2.7) ¥½Â¼êx. K3ü
H : C \ 0 → Λ Ú^²L −λ/3 ©l 0 Ú ∞ ü4 Γ ⊂ C \ 0 ¦(1) XJ α ∈ Γint, K ∂∆fα TвL 2 .: π/2 Ú −π/2.
(2) XJ α ∈ Γext, K ∂∆fα TвL 2 .: H(α) Ú −H(α).
(3) XJ α ∈ Γ \ −λ/3, K ∂∆fα TвL 4 .: ±π/2 Ú ±H(α).
(4) XJ α = −λ/3, K ∂∆fα TвL 2 ê 3 .: π/2 Ú −π/2.
§1.3. ÎÒ(² 9
§1.2.4 ng tableaux ¢y½n
zäkØëÏ Julia 8ngõª f ѱ½Â N ng tableau
½IP: (marked grid), §´÷v,^ M(j, k) ∈ 0, 1 | j, k ≥0 j + k ≤ N. Branner Ú Hubbard [11] y²ngõgIP:7L÷v
K. Ó, ±½Â N ÄIP: M = M(j, k) ∈ 0, 1 | j, k ≥0 j + k ≤ N [11, 19].
XJngõª f tableau ÄIP: M Ó, K¡ M ù
ngõª f ¤¢y. Branner Ú Hubbard y²e¡½n.
½n 1.2.16 (Tableaux ¢y, [11]). ?ÛIP:ѱdngõª¤¢
y.
ÄäkØëÏ Julia 8õª f 3ááÚ Ω(f) þ. XJPò f
3 Ω(f) þ³ëÏ©|l¤:ûm T , K f : Ω(f)→ Ω(f) p
üXä T þÄåXÚ F : T → T ( [30, 18]). d F : T → T ÄåXÚ
±Ä/½Âaõªä (½Â 6.2.1).
318Ù¥, ·ò^äþÄåXÚѽn 1.2.16 #y². ÙÌg
´: k`²?ÛIP:±dÓ “Д 3 g children N¢y,
ù 3 g children N±òÿNÝu 3 aõªä. duz 3 g
aõªäѱd 3 gõª¤¢y (½n 6.2.2), l ½n 1.2.16 y.
§1.2.5 McMullen NááÚ>. Hausdorff ê
·3Ø©ÙÑëêªu 0 , McMullen NááÚ
>. Hausdorff êìCúª.
½n 1.2.17. - n ≥ 3, bëê λ ¿©¦ McMullen N fλ(z) = zn + λ/zn
Julia 8´ Cantor ±. K fλ ááÚ>. ∂Bλ Hausdorff
ê´
dimH(∂Bλ) = 1 +|λ|2
log n+O(|λ|3). (1.2.9)
AO/, XJ n 6= 4, Kp O(|λ|3) ±d O(|λ|4) O.
§1.3 ÎÒ(²
·Ñ©ÎÒ(². P
• E²¡ C;
10 1Ù XØ
• Riemann ¥¡ (¡*¿E²¡) C = C ∪ ∞;• þE²¡ H := z ∈ C : Im(z) > 0;• % u a ∈ C » r > 0 î¼ D(a, r) := z ∈ C : |z − a| < r;• % u 0 » r Dr := D(0, r);
• % u 0 » r ± Tr;• ü D := D1;
• ü ± S1 := ∂D;
• A(r1, r2) z ∈ C : r1 < |z| < r2;• R ¢, Z ê8, N∗ ê8.
éu C ¥?Ûüm8 A Ú B, ·^ A ⊂⊂ B L« A ⊂ B. X, Y C (½
C) þ;f8 x ∈ X, ½Â x Y ¥¡ (½î¼) ål d(x, Y ) = infy∈Y |x− y|.8Ü X Ú Y m¥¡ (½î¼) ål d(X, Y ) = infx∈X,y∈Y |x− y|, Ù¥ | · | L«¥¡ (½î¼) Ýþ. 8Ü X ¥¡ (½î¼) »½Â diam(X) = supx1,x2∈X |x1 − x2|.©¥ÑyëêmÚÜ© Julia 8ã´æ^ A. Cheritat JøÑ
. 'u Julia 8ãëw©z [12, 55] Ú [53, N¹ H].
1Ù E©ÛÚEÄåXÚ£ý
ùÙ·OE©ÛÚEÄåXÚ¥ÄVgÚ½n. Ì)[/
Nüd½Â, ÿ Riemann N½n, Koebe ½n, Fatou 8, Julia 8½
Â, kn¼êÄåXÚ©a.
§2.1 EA, [/N, ÿ Riemann N½n
E¼ê f(z) ´« Ω ⊂ C þëY¼ê, = f ¢Ü u ÚJÜ v
¢Cþ x, y ¼êþäkëY ê. d du = uxdx + uydy, dv = vxdx + vydy,
f äkE©/ª: df = du + idv = (ux + ivx)dx + (uy + ivy)dy. P dz =
dx+ idy, dz = dx− idy, d dx = (dz + dz)/2, dy = (dz − dz)/(2i), ·±ò df U
df = fzdz + fzdz, Ù¥
fz = (ux + vy)/2 + i(vx − uy)/2,
fz = (ux − vy)/2 + i(uy + vx)/2.(2.1.1)
½Â 2.1.1. ·¡ (2.1.1) ª¥ fz 9 fz f ü/ª ê.
½Â 2.1.2. f ´« Ω ⊂ C SëY~¼ê z0 ∈ Ω, e
fz(z0) 6= 0, ½Â f 3 z0 EA (complex dilatation ) µf (z0) = fz(z0)/fz(z0).
âEA½Â, e f /N, K µf = 0.
·Äk0[/NAÛ½Â, dk7kÚ?/£. \
[SNëw©z [44] 1nÙ.
P Q C ¥ Jordan « (>.ü4), 3Ù>.þUgÀ½
ØÓo: z1, z2, z3, z4, K Q ëÓùo:¤ÿÀo>/, P Q(z1, z2, z3, z4). ù
po:¡o>/º:.
½Â 2.1.3. éu*¿E²¡ C ¥?ÛÿÀo>/ Q(z1, z2, z3, z4), Ñ3¢
ê a > 0, b > 0,¦ Q(z1, z2, z3, z4)/duÝ/ R(0, a, a+bi, bi). ùÝ/>
' a/bdÿÀo>/ Q(z1, z2, z3, z4)û½. Ù¥ a/b¡ÿÀo>/ Q(z1, z2, z3, z4)
/ (conformal modulus ), ¡, P Mod(Q(z1, z2, z3, z4)). 3²(ÿÀo>
/oº:ÚgS, Q(z1, z2, z3, z4)P Q, Mod(Q(z1, z2, z3, z4))PMod(Q).
½Â 2.1.4 (AÛ½Â). f : D → G ´ C ¥« D G Ó, e
3~ê K ≥ 1, ¦éu D ¥zÿÀo>/ Q, Q ⊂ D, þk Mod(f(Q)) ≤K Mod(Q),K¡ f ´ K [/N (quasiconformal mapping ),P K–qcN.
11
12 1Ù E©ÛÚEÄåXÚ£ý
N´Ñ, f : D → G ´ K–qc N¿^´µ
Mod(Q)/K ≤ Mod(f(Q)) ≤ K Mod(Q), Ù¥ Q ⊂ D.
Ñ[/N©Û½Â. d·kÚ\¼êäk ACL 5Vg.
½Â 2.1.5. f ´½Â3 D ⊂ C SEëY¼ê. XJéu?¿Ý/ R =
x + iy : a < x < b, c < y < d, R ⊂ D\∞, f−1(∞), N f(x, y) éA¤k½
x ∈ (a, b) Ñ´ y ýéëY¼ê, éA¤k½ y ∈ (c, d) Ñ´ x ýéëY¼
ê. K¡¼ê f 3 D SãþýéëY, ¡ f äk ACL 5.
Ún 2.1.6 ([44, p. 77-79]). f : D → G « D G [/N, K f 3
D Säk ACL 5, Ï f A??k ê, A?? .
½Â 2.1.7 (©Û½Â, [2, p. 16]). D,G þ C ¥«, f : D → G Ó
, XJ f ÷ve^µ
( 1 ) f 3 D þ´ ACL ¶
( 2 ) |µf (z)| ≤ (K − 1)/(K + 1) 3 D SA??¤á.
K¡ f : D → G ´ K–qc N.
[/N©Û½ÂÚA۽´d ([2, p. 20]).
íØ 2.1.8. XJ[/N f ÷v µf = 0 a.e., K f /N.
y². dÚn 2.1.6 f ÷v ACL 5, 2â µf = 0 a.e., éì[/N
©Û½Â K = 1. Ï f 1–qc N, = f /N.
½Â 2.1.9. éu« ΩþE¼ê µ,XJ µ´ Lebesgueÿ÷v ‖µ‖∞ <
1, K¡ µ Ω þ Beltrami Xê. § fz = µ(z)fz K¡ Beltrami §.
éu½ Ω þ Beltrami Xê µ Ú ν, µ Ú ν 3 Ω þA??, ·ò
µ Ú ν wÓ Beltrami Xê, P µ = ν. XJ f ´[/N, KÙEA
µf = fz/fz ´ Beltrami Xê.
éu½ Beltrami Xê, ´Ä½3[/N±ÙEAQ? e
¡ÿRiemannN½n (Measurable Riemann Mapping Theorem) Ñ£.
½n 2.1.10 (Ahlfors and Bers, [3]). - µ C þ Beltrami Xê, z1, z2, z3
C þØÓn:, w1, w2, w3 C þØÓn:. K3[/N
f : C→ C, ¦ fz = µ(z)fz 3 C þA??¤á, k f(zn) = wn, n = 1, 2, 3.
§2.2. RIEMANN N½n, KOEBE ½n, 13
e ϕ ψ äkÓ Beltrami Xêü[/N, K ϕ Ú ψ =
EÜ/N, = ϕ = φ ψ, Ù¥ φ /N.
éõ©z¥ò Beltrami § L2 2ÂÓ)½Â[/N. ½n 2.1.10 ¥
ÿ¼ê µ(z) ¡[/N f EA, K(f) = (1 + ‖µf‖)/(1−‖µf‖) ¡ f
() û, Ù¥ ‖µf‖ L« f Beltrami Xê3 L∞ m¥ê.
½n 2.1.11 (Ahlfors and Bers, [3]). µ = µ(z, t) ´½Â3 C×D þ (½ C× [0, 1]
þ) k.ÿ¼ê, |µ(z, t)| ≤ k < 1 éA¤k z ∈ C Úé?¿ t ∈ D (½
[0, 1])Ѥá, Ù¥ k ~ê . d, b½éA¤k½ z ∈ C, µ(z, t) ´ t ∈ D)Û¼ê (½ [0, 1] þëY¼ê). qéu?¿½ t, ω = f(z, t) ´ C g[/N, ÙEA µ(z, t), ± 0, 1 ∞ ØÄ. Kéu?¿½ z ∈ C,
ω = f(z, t) ´ t ∈ D )Û¼ê (½ [0, 1] þëY¼ê) .
§2.2 Riemann N½n, Koebe ½n,
Riemann N½n´ü¼êØ¥~½n.
½n 2.2.1 (Riemann N½n). Ω ⊂ C ´üëÏ«, >.õu:.
q z0 ∈ Ω ´?¿½:, K3/N f : Ω→ D, ò Ω Nü . X
J¦ f ′(z0) = 0 f ′(z0) > 0, Kù/N f ´.
½Â 2.2.2. X ´ C þk.8Ü, XJé?¿ ε > 0, 3 δ > 0 ¦é X
¥?¿ü: a, b, |a − b| < δ, Ò½3 X ëÏf8 E ÷v: a, b ∈ E
diam(E) < ε, K·¡ X ´ÛÜëÏ.
½n 2.2.3 (Caratheodory òÿ½n). f : D→ Ω ´/N, K f ±ëYò
ÿ>.= Ω ´ÛÜëÏ.
e¡Ñ3ü¼êØ¥~k^ ½n.
½n 2.2.4 (Koebe ½n, [60, p. 9]). f : D → C ´ü)Û¼ê, Ké
?¿ z ∈ D, k
|f ′(0)| |z|(1 + |z|)2
≤ |f(z)− f(0)| ≤ |f ′(0)| |z|(1− |z|)2
;
|f ′(0)| 1− |z|(1 + |z|)3
≤ |f ′(z)| ≤ |f ′(0)| 1 + |z|(1− |z|)3
.
14 1Ù E©ÛÚEÄåXÚ£ý
½Â 2.2.5. A C þëÏ«, XJ A /du A1,R = z ∈ C : 1 <
|z| < R, K¡ A ÿÀ¿½Â A ½/
mod(A) =1
2πlogR.
e A /du C \ 0 ½ D \ 0, K½Â§ ∞.
½n 2.2.6 (Grotzsch ت). A,A1, A2 þëÏ, A1 ∩ A2 = ∅, Ai ⊂ A
C \ A ü©|©Oá\ C \ Ai ü©|¥, i = 1, 2. Kk
mod(A1) + mod(A2) ≤ mod(A).
§2.3 ±Ï:, Fatou 8, Julia 8
·ÄX¼ê f : X → X ÄåXÚ, Ù¥ X o Riemann ¥¡ (k
n¼ê/) oE²¡ (¼ê/). e f : C → C knN, K f
L« P (z)/Q(z), Ù¥ P Ú Q põª, f NÝ (degree)½Â
deg(f) = maxdeg(P ), deg(Q).Ä: z0 3 f : X → X eS: z0 7→ z1 = f(z0) 7→ z2 = f 2(z0) 7→ · · · , ¡S
f n(z)n≥0 z0 3 f ec; (forward orbit), P O+f (z0). e3
ê p ¦ f p(z0) = z0, K¡ z0 f ±Ï: (periodic point), ¤k p ¥ö¡
z0 ±Ï (period). d z0 ;P z0, z1, · · · , zp−1, ¡d; f ±Ï;
(periodic orbit ½ cycle). w,±Ï;Sz:Ñ´±Ï:¿äkÓ±Ï. A
O/, e p = 1, K¡ z0 f ØÄ: (fixed point).
½Â 2.3.1. éu±Ï p ±Ï; f : z0 7→ z1 7→ · · · 7→ zp−1 7→ zp = z0,
½ÂEê λ = (f p)′(zi) =∏p−1
i=0 f′(zi) T;¦f (multiplier). ±Ï;¥:
¹ ∞ , d·½ÂT;¦f λ = 1/(f p)′(∞), Ù¥ (f p)′(∞) ½Â4
limz→∞
(f p)′(z)([6], p.41).
±Ï;¦fXJ©O÷v |λ| < 1!|λ| > 1 Ú |λ| = 1, K¡ù;´á
5 (attracting)!½5 (repelling)Ú¥5 (indifferent). AO/, λ = 0 ¡T;
´á5 (superattracting), 0 < |λ| < 1 K¡T;´AÛá5 (geometrically
attracting). λ = e2πiθ θ knê, ¡T±Ï;´kn¥5 (rational indiffer-
ent), XJ ∀n, f n ÑØðN, Kkn¥5±Ï;¡´Ô (parabolic).
A θ Ãnê, K¡T;¡Ãn¥5 (irrational indifferent)±Ï;.
§2.3. ±Ï:, FATOU 8, JULIA 8 15
- f Ú g ´ü/ÝX¼ê,=3MobiusC h,¦ hf h−1 =
g. 1e f äk±Ï; C, K g äkA±Ï; h(C), â½Â§äk
Ó¦f. ÏdX¼ê3/Ý^e±Ï;5ØC.
½Â 2.3.2. F ´lÝþm X Ýþm Y ¼êx, XJéu X ¥
z: x ±9 F ¥?ÛáS, Ñ3ÙfS±9 x U ¦TfS
3 U þ´Âñ, K¡¼êx F X þ5x (normal family).
½Â 2.3.3. f : X → X Riemann ¥¡½E²¡gXN, ½Â¤k
¦ f S¼êx f nn≥0 5«¿ f Fatou 8, P F (f), Ù8
X \ F (f) ½Â f Julia 8, P J(f).
d½Â Fatou 8m8, Julia 8K48.
½Â 2.3.4. F Ýþm (X1, d1)Ýþm (X2, d2)ëY¼êx, ∀ε > 0,X
Jéu X1 ¥: x0, 3ê δ, ¦ d1(x, x0) < δ , k d2(f(x), f(x0)) < ε
é F ¥?¿¼ê f Ѥá, Kd·¡¼êx F 3 x0 ù:´ÝëY
(equicontinuous). XJéu X1 ¥z:ѱéA δ,@o·Ò¡¼
êx F 3 X1 þ´ÝëY.
â Arzela-Ascoli ½n ( [1]), kn¼êx3«þ55duT¼êx
ÝëY5, ¤±: z áu f Fatou 8du3¹ z m U , ¦ f S
¼êS f nn≥0 3 U þ´ÝëY.
âÝëY5½Â, éu?¿ ε > 0, e z ∈ F (f), K3 δ > 0, ¦
é?¿g,ê n, ± z %, δ »m3 f n ^eá3»Ø
L ε ¥. ùL²éu Fatou 8¥:5`, T:¿©3 f ?¿gS
^e´¿©. Julia 8¥:3 f SeKwéЩ'¯a, =e
z ∈ J(f), Kéu z ?¿m U , 73 U ¥ü:, ¦ùü:3 f Se¥
yØÓÄåXÚ5.
â Montel ½n, X¼êx F = fα : D → C, α ∈ A ´« D þ5
x= F 3 D þ´ÛÜk..
½Â 2.3.5. - U F (f) ëÏ©|, K U ¡ f Fatou©| (Fatou
component). e3ê p ≥ 1, ¦ f p(U) = U , K¡ U f ±Ï Fatou ©
| (periodic Fatou component); XJ3 n ≥ 0, ¦ f n(U) ´±Ï Fatou ©|, K¡
U ´ª±Ï (eventually periodic). XJ U Ø´ª±Ï, ·¡ U f
i (wandering domain).
16 1Ù E©ÛÚEÄåXÚ£ý
½n 2.3.6 ([71]). kn¼êvki.
ù½n`²?ÛNÝu 1kn¼ê Fatou©|ª7N±Ï
Fatou ©|¥, Ï éukn¼ê Fatou 8©aIÄ±Ï Fatou ©|Ò1.
éukn¼ê±Ï Fatou ©|, ·kXe©a½n.
½n 2.3.7 ([51]). U kn¼ê f ±Ï p Fatou©|,Ù¥ deg(f) ≥2, K U =XeÊ«/ (ã2.1) :
() áÚ Ô Siegel Herman
ã 2.1 Ê« Fatou ©|.
(AB) áÚ, U ¹ f ±Ï p á5±Ï: z0, 3 U ?Û;f8
þk f np(z) Âñ: z0 (n→∞)¶
(SAB) áÚ, (AB)aq, =òá5±Ï:Uá5±Ï:¶
(PB) Ô, 3 U >.þ3Ô±Ï: z0, ¦ f p(z0) = z0, f p 3 z0
¦fu 1, 3 U ?Û;f8þk f np(z) Âñ: z0 (n→∞)¶
(SD) Siegel,3/N ψ : U → DÚÃnê θ,¦ ψf pψ−1(z) =
e2πiθz, = f p 3 U þ^/Ýuü þÃn^=¶
(HR) Herman , (SD)aq, =Iò D ¤ A(r, 1) = z : r < |z| < 1,Ù¥ 0 < r < 1.
5¿¼ê±ki [5], ±Ï Fatou ©|©aØÓukn¼ê. 'X
¼êvk Herman ±k Baker .
k' Julia 8Ú Fatou 8Ù§5IOë©z [6, 13, 53] ½ö [8,
28, 40, 45, 68].
§2.4 Ak^Ún
éu?Ûf8 A ⊂ C, A î¼»½Â Diam(A) = supx,y∈A |x − y|. ?^Jordan C, TþüØÓ:ò§©¤üã. ^ I L«äkáî¼»
@ã, éu½~ê k, XJ Diam(I)/|x − y| ≤ k é C þ¤kØÓ x Ú y ¤á,
§2.4. Ak^Ún 17
K·¡ C ´ k–k.ò=. XJ^ Jordan ´ k–k.ò=, K§´ K–[,
Ù¥~ê K =6u k, ½, ( [43, p.100]).
·^ |I| L«©¡1w½öl I î¼Ý. 5¿ Tr = z : |z| = r, Ù¥r > 0.
Ún 2.4.1. b f : D → C ´ü. éu? 0 < r < 1, K f(Tr) ´[ f(Tr) ûk=6u r þ..
y². é?ÛØÓ x, y ∈ f(Tr), - I, J f(Tr) \ x, y ü¦ |I| ≤ |J | ëÏ©|. ·äó3Ø6u f, xÚ y ~ê C(r) > 0,¦ |I| ≤ C(r)|x−y|.
- L = [x, y] ë x Ú y ã, K |L| = |x − y|. e¡?Øò©¤ü«¹:
L ⊂ D Ú L 6⊂ D, Ù¥ D = f(D). - x1, y1 ∈ D Ú I1 ⊂ D ©O x, y, I c. X
J L ⊂ D, P L1 = f−1(L), §´ D ¥^1w. ½ r0 ∈ (r, 1), ·^ Ci(r0)
L«=6u r0 ~ê (Ï =6u r). XJ L1 ⊂ Dr0 , Kâ Koebe ½nk
|L| ≥ C1(r0)|x1 − y1| |I| ≤ C2(r0)|I1| ≤ 2C2(r0)|x1 − y1|. ùL² |I| ≤ C3(r0)|L|. XJ L1 6⊂ Dr0 , - L2 L1 ∩ Dr0 ¹ x1 ©|. â Koebe ½n, |I| ≤ C2(r0)|I1|,|I1| ≤ C4(r0)|L2| ±9 |L| ≥ C5(r0)|L2|. Ïd |I| ≤ C6(r0)|L|. XJ L 6⊂ D, ?Øa
qu L1 6⊂ Dr0 ¹. Úny..
íØ 2.4.2. - D Jordan «. b γ ⊂ D ´[ f : D → C ´ü. K f(γ) ´ K–[, Ù¥ K = K(D, γ) ´=6u D Ú γ Ø6u
f ~ê.
Ún 2.4.3. - 0 < x < 1 Ãnê, K8Ü M = e2πkxi | k ∈ N 3ü ±þÈ.
y². b3 S1 þml M Ø. À¥Ù¥^, P I, ¦
I ∩M = ∅. ùL²é¤k k ≥ 1, Ik = e−2πkxiz | z ∈ I M Ø. Ï I ´
, K73 k ∈ N ¦ I = Ik. l x ´knê. ùgñL²Ún¤á.
1nÙ Julia 8 Cantor ±kn¼ê
ékn¼ê Julia8ÿÀ5ïÄ´EÄåXÚ¥¯K.éõö,
'X Avila, Buff, Cheritat, Branner, Douady, Hubbard, Lyubich, Pertersen, Shishikura,
Yoccoz, Zakeri ÑédÑïÄ [4, 11, 38, 46, 57, 58, 66]. ézNÝØu 2
õª, ©z [63] ¥®²y²ØêõëÏ©|, Ù§ Julia 8ëÏ©|Ñ
´ü:. éukn¼ê, Julia 8±ÑyE,ÿÀ. Pilgrim Ú Tan y²: XJ
V (/, AÛk) kn¼ê Julia 8Ø´ëÏ, KØkõ©|Ú§
êc±, Julia ©|o´ü:o´ Jordan [59, ½n 1.2]. ù
Ù¥, ·òÄakn¼ê, Ù Julia 8äküÿÀ: z Julia 8©|Ñ´
Jordan .
§3.1 Úó
XJ Riemann ¥¡ C f8Óu C × S1, Ù¥ C Cantor n©8 S1
ü ±,KTf8¡´ Cantor ± (Cantor circles). 1 Julia8 Cantor
±~f´d McMullen uy ( [49, §7]). ¦y²XJ f(z) = z2 + λ/z3 ¥ λ v
, K f Julia 8´ Cantor ±. , õÖör5¿å8¥3e¡¼
êx, y3Ñ¡ McMullen N:
fλ(z) = zm + λ/zl, (3.1.1)
Ù¥ l,m ≥ 2 λ ∈ C \ 0 ( [24, 69, 61] Úp¡ë©z). λ éÿ, ùx
AÏkn¼ê±w¤´üõª f0(z) = zm 6Ä. ®² 1/l + 1/m < 1
, 3:B M, ¡´ McMullen , ¦ λ ∈ M , ¼ê fλ
Julia 8´ Cantor ± ( [49, §7] Ú [24, §3]).
y3kXeng,¯K: (1) Ø McMullen N, ´Ä3Ù§kn¼
êÙ Julia 8 Cantor ±? (2) XJ1¯KY´½, @§´o?
ó, ·UÄéѧLª? (3) ·UÄ3½§ÝþéѤk Julia 8
Cantor ±kn¼ê? ùÙòÑù¯K½£.
d[/Ãâ,ò fλ ¹∞áÚ6Ä¡AÛáÚ,·±
éõ#kn¼ê. ½Ù¥,@oùN3 Riemann¥¡þ fλ Ø´ÿ
ÀÝ. §3éA Julia8þ´ÿÀÝ. AO/, gε,λ(z) = 1z(zm+ε) 1
z+λ/zl
´~f, Ù¥ 1/l+ 1/m < 1 ε, λ ∈ C \ 0 Ñ¿©. , , Ïùa.kn
¼ê±w¤´3ý McMullen N Fatou 8þÃâ, ¤±3
18
§3.1. Úó 19
þ§´±w¤´ McMullen N, ùØ´·é. Ïd~k¯
K´, éÙ§¦ Julia 8´ Cantor ±kn¼ê, §3 Julia 8þØÿÀ
Ýu?Û McMullen N.
®² “þ” McMullen NØÓkn¼ê´3 ( [35, §§1,2]). ù
Ù¥, ·òÑùa.kn¼êäNLª, Ø=¹ [35] ¥?ع,
3 “þ” ¹¤k Julia 8 Cantor ±kn¼ê (½n 3.1.2). -
p ∈ 0, 1, n ≥ 2 ê d1, · · · , dn n ¦∑n
i=11di< 1 ê. ·½Â
fp,d1,··· ,dn(z) = z(−1)n−pd1
n−1∏i=1
(zdi+di+1 − adi+di+1
i )(−1)n−i−p , (3.1.2)
Ù¥ a1, · · · , an−1 n − 1 ÷v 0 < |a1| < · · · < |an−1| < 1 Eê. AO/, XJ
n = 2, K f1,d1,d2(z) = zd2 − ad1+d21 /zd1 ´éõö®²ïÄ McMullen N. d
, f0,d1,d2(z) = zd1/(zd1+d2 − ad1+d21 ) /Ýu McMullen N z 7→ zd1 + λ/zd2 , Ù¥
λ 6= 0. ¼ê fp,d1,··· ,dn 3 0Ú∞?NÝ©O´ d1 Ú dn deg(fp,d1,··· ,dn) =∑n
i=1 di.
éx (3.1.2) ¥z, N´y 0 Ú ∞ áu fp,d1,··· ,dn Fatou 8. - D0 Ú D∞
©O¹ 0 Ú ∞ Fatou ©|. y3ke¡o«¹:
(1) XJ p = 1 n Ûê, K f(D0) = D0 f(D∞) = D∞;
(2) XJ p = 1 n óê, K f(D0) = D∞ f(D∞) = D∞;
(3) XJ p = 0 n Ûê, K f(D0) = D∞ f(D∞) = D0;
(4) XJ p = 0 n óê, K f(D0) = D0 f(D∞) = D0.
Äk·òéÑ (3.1.2) ¥·ëê ai, Ù¥ 1 ≤ i ≤ n− 1, ¦þ¡o«¹
¥ fp,d1,··· ,dn Julia 8Ñ´ Cantor ±. - ξ =∑n
i=11di K ≥ 3 d1, · · · , dn ¥
ê.
½n 3.1.1. XJ |an−1| = (s1K−2)1/dn, é 1 ≤ i ≤ n − 2 k |ai| = (s1K
−5)1/di+1
|ai+1|, Ù¥ s1 > 0 ¿©, K f1,d1,··· ,dn Julia 8´ Cantor ±. XJ |an−1| =
(s1/dn+(1−ξ)/30 )1/dn éu 1 ≤ i ≤ n−2 k |ai| = (s
1+1/dn+2(1−ξ)/30 )1/di+1|ai+1|, Ù¥ s0 > 0
¿©, K f1,d1,··· ,dn Julia 8´ Cantor ±.
3,«§Ýþ, ½n 3.1.1 L«·®²éxäkëê s1 Ú s0 kn¼ê,
§ Julia 8 Cantor ±. Ï s1 Ú s0 ±?¿, ùkn¼êw¤´ zdn
Ú z−dn 6Ä (éAu p = 1 Ú 0). §3.2 ¥Ñëê s1 Ú s0 äN
( (3.2.1) Ú (3.2.2)). d, ·òy²XJ n ≥ 3, Kz fp,d1,··· ,dn 3 Julia 8þØ´
ÿÀÝu McMullen N (½n 3.2.6). ùL²·éakn¼êäNL
ª¦§ Julia 8´ Cantor ±, § McMullen N3þ´ØÓ.
20 1nÙ Julia 8 Cantor ±kn¼ê
5¿XJkn¼ê f Julia 8 J(f) ´ Cantor ±, Ï z Julia ©
|Ñ´ Jordan , @o3 J(f) þBØ3.: (Ún 3.3.1). ùL² f z±
Ï Fatou ©|o´á5o´Ô. ¯¢þ, ·kXe½n
½n 3.1.2. f kn¼ê, Ù Julia 8 Cantor ±. K3 p ∈ 0, 1, ê n ≥ 2 ±9÷v
∑ni=1
1di< 1 d1, · · · , dn ¦ f Ú fp,d1,··· ,dn 3§éA Julia
8þ´ÿÀÝ, fp,d1,··· ,dn ëê ai d½n 3.1.1 (½, Ù¥ 1 ≤ i ≤ n− 1.
â½n 3.1.2· “A”é¤k Julia8 Cantor±kn¼ê. X
Jkn¼ê.:Ñá5±Ï;¤áÚ, KTkn¼ê¡´V. éu
fp,d1,··· ,dn Julia 8©|K5, ±y²XJ fp,d1,··· ,dn ´V, K fp,d1,··· ,dn z
Julia 8©|Ñ´[ (íØ 3.3.3).
XJ λ v, K fλ ´V ( [24]). y3·EVkn¼ê¦
§ Julia 8´ Cantor ±. - m,n ≥ 2 ü÷v 1/m + 1/n < 1 ê
λ ∈ C \ 0, ·½Â
Pλ(z) =1n((1 + z)n − 1) + λm+nzm+n
1− λm+nzm+n. (3.1.3)
y 0 ´ Pλ ¦f 1 ÔØÄ:. ·k
½n 3.1.3. XJ 0 < |λ| ≤ 1/(210mn3), K Pλ ´VÙ Julia 8 Cantor
±.
ɽn 3.1.1 éu, ·±EõVkn¼ê¦§ Julia 8´
Cantor ±. üå, éz n ≥ 2 ±9 1 ≤ i ≤ n, ·=Ĺ di = n+ 1. é
z n ≥ 2, ·½Â
Pn(z) = An(n+ 1)z(−1)n+1(n+1)
nzn+1 + 1
n−1∏i=1
(z2n+2 − b2n+2i )(−1)i−1
+Bn, (3.1.4)
Ù¥ b1, · · · , bn−1 n− 1 ÷v 1 > |b1| > · · · > |bn−1| > 0 Eê
An =1
1 + (2n+ 2)Cn
n−1∏i=1
(1−b2n+2i )(−1)i , Bn =
(2n+ 2)Cn1 + (2n+ 2)Cn
Cn =n−1∑i=1
(−1)i−1b2n+2i
1− b2n+2i
.
(3.1.5)
ùp An Ú Bn y Pn(1) = 1 P ′n(1) = 1. =, 1 Pn ¦f 1 Ô
ØÄ: (Ún 3.7.1).
½n 3.1.4. éz n ≥ 2 Ú 1 ≤ i ≤ n− 1, XJ |bi| = si, Ù¥ 0 < s ≤ 1/(25n2),
K Pn ´VÙ Julia 8 Cantor ±.
§3.2. .: ±9V/ 21
±wÑéz n ≥ 2, Pn 3Ù Julia 8þÄåXÚÝu fp,n+1,··· ,n+1 3
Julia 8þÄåXÚ (p = 1). §3 Fatou 8þÄåXÚ«O´ fp,n+1,··· ,n+1
3 ∞ áÚ¤ Pn Ô.
ùÙ´ùSü: 31 §3.2¥,·O,y²½n 3.1.1. 3 §3.3¥,
·y²½n 3.1.2. 3 §3.4 ¥, ·òOÿÀd Cantor ±kn¼êÝa
ê8¿3 §3.5¥Ñùê8þe.O.3 §3.6¥,·y² λvÿ,
Pλ Julia 8´ Cantor ±, ùBy²½n 3.1.3. ·ò3 §3.7 ¥y²½n
3.1.4 ¿r'Ún3 §3.8.
§3.2 .: ±9V/
·ÄkÑÄÚk^O.
Ún 3.2.1. - n ≥ 2 ê, a ∈ C \ 0 0 < ε < 1/2.
(1) XJ |z − a| ≤ ε|a|, K |zn − an| ≤ ((1 + ε)n − 1) |a|n;
(2) XJ |zn − an| ≤ ε|a|n, K |a/z|n < 1 + 2ε 3, 1 ≤ j ≤ n ¦ |z −ae2πij/n| < ε|a|;
(3) XJ 0 < ε < 1/n, K nε < (1 + ε)n − 1 < 3nε nε/3 < 1− (1− ε)n < nε.
y². (1) - z = a(1 + reiθ), Ù¥ 0 ≤ r ≤ ε 0 ≤ θ < 2π, K
|zn − an| = |(1 + reiθ)n − 1| · |a|n ≤ ((1 + ε)n − 1) |a|n.
(2) XJ 0 < ε < 1/2, K |a/z|n ≤ 1/(1− ε) < 1 + 2ε, u´ (2) ¥1ت¤
á. éu1(Ø, - zn = an(1 + reiθ), Ù¥ 0 ≤ r ≤ ε 0 ≤ θ < 2π, K3,
1 ≤ j ≤ n ¦ z = ae2πij/n(1 + reiθ)1/n. d, XJ n ≥ 2, K
|z − ae2πij/n| = |(1 + reiθ)1/n − 1| · |a| ≤ ((1 + ε)1/n − 1) · |a| < ε|a|.
(3) é x 7→ xn3«m [1, 1 + ε] Ú [1− ε, 1] þ©O|^.KF¥½n.
½ n ≥ 2 - d1, · · · , dn ≥ 2 n ¦ ξ =∑n
i=11di< 1 ê. - K ≥ 3
d1, · · · , dn ¥ê. - u1 = s1K−5 v1 = s1K
−2, Ù¥
0 < s1 ≤ minK−5ξ/(1−ξ), K5−2K < 1. (3.2.1)
- u0 = s1+1/dn+2(1−ξ)/30 , v0 = s
1/dn+(1−ξ)/30 , Ù¥
0 < s0 ≤ min2−(1−ξ)−1(1+1/dn−2ξ/3)−1
, (4K)−3/(1−ξ), K−2K(1+1/dn+2(1−ξ)/3)−1 < 1. (3.2.2)
22 1nÙ Julia 8 Cantor ±kn¼ê
éu p ∈ 0, 1,- |an−1,p| = v1/dnp |ai,p| = u
1/di+1p |ai+1,p|¼êx fp,d1,··· ,dn ¥ n− 1
ëê, Ù¥ 1 ≤ i ≤ n− 2. Ï3¹e, éu p = 0 Ú p = 1 ùü«¹±Ó
?1?Ø, Bå, ØuÚå· ¹e, ·^ s, u, v Ú ai ©O5L«
sp, up, vp Ú ai,p, Ù¥ 1 ≤ i ≤ n− 1.
Ún 3.2.2. (1) u2/K ≤ K−4.
(2) XJ 1 ≤ j ≤ i ≤ n− 1, K |aj/ai| ≤ ui−jK .
(3) XJ p = 1, K (3a) (s/|a1|)d1 < su/(2v) = sK−3/2 (3b) (|a1|/s)d1v/2 > K.
(4) XJ p = 0, K (4a) 2Ku/v < s 1/(2Kv) > (2/s)1/dn; (4b) (s/|a1|)d1 <sv/2 < u1/2/2 (4c) (|a1|/s)d1u/(2v) > (2/s)1/dn.
y². (1)â (3.2.1)Ú (3.2.2),·k s1 ≤ K5−2K Ú s0 ≤ K−2K(1+1/dn+2(1−ξ)/3)−1.
ùL² u2/K1 = (s1K
−5)2/K ≤ K−4 u2/K0 ≤ K−4.
(2) XJ j = i, K(Ø´w,. b 1 ≤ j < i ≤ n − 1, Ïéu 1 ≤ i ≤ n k
K ≥ di, K
|aj/ai| = u1
dj+1+···+ 1
di ≤ ui−jK .
ùÒy² (2).
(3)XJ p = 1,K u = sK−5 v = sK−2. Ï s ≤ K−5ξ/(1−ξ),·k s1−ξK5ξ ≤ 1,
u´
s1− 1
d1 s−( 1
d2+···+ 1
dn)K
5( 1d2
+···+ 1dn−1
)+ 2dn 2
1d1K
3d1 < 1.
Ï
|a1| = u1d2
+···+ 1dn−1 v
1dn = s
1d2
+···+ 1dn /K
5( 1d2
+···+ 1dn−1
)+ 2dn ,
þªdu s1− 1
d1 21d1K
3d1 /|a1| < 1. Ïd·k (s/|a1|)d1 < su/(2v) = sK−3/2 (3a)
y². d, Ï (|a1|/s)d1 > 2K3/s = 2K/v, (3b) ±d (3a) íÑ.
(4) XJ p = 0, K u = s1+1/dn+2(1−ξ)/3, v = s1/dn+(1−ξ)/3. â (3.2.2), ·
4Ks(1−ξ)/3 ≤ 1, ù¿X 2Ku/v = 2Ks1+(1−ξ)/3 < s. 5¿ 21+1/dnKs(1−ξ)/3 < 1, ù
du 1/(2Kv) > (2/s)1/dn . ù(å (4a) y².
d (3.2.2), ·
1 ≥ 2s(1−ξ)(1+1/dn−2ξ/3) > 21d1 s(1−ξ)(1+1/dn−2ξ/3)
= 21d1 s
1− 1d1 /s
( 1d2
+···+ 1dn−1
)+ 1dn
( 1d1
+···+ 1dn
)+2ξ(1−ξ)
3
> 21d1 s
1− 1d1 /s
( 1d2
+···+ 1dn−1
)+ 1dn
( 1d1
+···+ 1dn
)+ 1−ξ3
( 1d1
+2( 1d2
+···+ 1dn−1
)+ 1dn
)
= s1− 1
d1 (2/v)1d1 /|a1|.
ù¿X (s/|a1|)d1 < sv/2 = u1/2s(1+1/dn)/2/2 < u1/2/2. Ïd (4b) ¤á.
§3.2. .: ±9V/ 23
(4c) y²aqu(4b). ·I5¿
1 ≥ 2s(1−ξ)(1+1/dn−2ξ/3) > 21d1
(1+ 1dn
)s(1−ξ)(1+1/dn−2ξ/3) > (s/|a1|)(2v/u)
1d1 (2/s)
1d1dn .
ùL² (|a1|/s)d1u/(2v) > (2/s)1/dn .
e5, üå, ·^ f L« fp,d1,··· ,dn . 5¿ 0 Ú ∞ ´ f ê d1 − 1
Ú dn − 1 .:, f NÝ´∑n
i=1 di. P Di = di + di+1, ·k 5 ≤ Di ≤ 2K,
Ù¥ 1 ≤ i ≤ n− 1. Ø 0 Ú ∞, f ∑n−1
i=1 Di .:´e¡§)
(−1)p zf ′(z)
f(z)=
n−1∑i=1
(−1)n−iDizDi
zDi − aDii+ (−1)nd1 = 0. (3.2.3)
éu 1 ≤ i ≤ n− 1, - CP i := wi,j = riai exp(πi2j−1Di
) : 1 ≤ j ≤ Di þ!á3± Tri|ai| þ Di :8Ü, Ù¥ ri = Di
√di/di+1. e¡ÚnL« f
∑n−1i=1 Di
gd .: “~C”⋃n−1i=1 CP i.
Ún 3.2.3. éz wi,j ∈ CP i, Ù¥ 1 ≤ i ≤ n− 1 1 ≤ j ≤ Di, 3 (3.2.3)
) wi,j, ¦ |wi,j − wi,j| < u2K |ai|. d, wi1,j1 = wi2,j2 = (i1, j1) = (i2, j2).
y². 5¿ (3.2.3) m>ªdu
(−1)n−i(
DizDi
zDi − aDii− di
)+Gi(z) = 0, (3.2.4)
Ù¥
Gi(z) =∑
1≤j≤n−1, j 6=i
(−1)n−jDjzDj
zDj − aDjj+ (−1)nd1 + (−1)n−idi. (3.2.5)
3 (3.2.4) ü>Ó¦± (zDi − aDii )/di+1, Ù¥ 1 ≤ i ≤ n− 1, ·k
(−1)n−i(zDi + diaDii /di+1) + (zDi − aDii )Gi(z)/di+1 = 0. (3.2.6)
- Ωi = z : |zDi + diaDii /di+1| ≤ ε |ai|Di, Ù¥ ε = u
2K 1 ≤ i ≤ n − 1. éz
z ∈ Ωi, ÏâÚn 3.2.2(1) ε ≤ K−4, ·k
K−1 < di/di+1 − ε ≤ |z/ai|Di ≤ di/di+1 + ε < K − 1 < K. (3.2.7)
ùL²
K−1 < |ai/z|Di < K l K−1 < |ai/z|5 < K. (3.2.8)
XJ 1 ≤ j < i Ú z ∈ Ωi, ·k
|aj/z|Di ≤ |ai/z|Di |ai−1/ai|Di < Ku1+di+1/di < 1. (3.2.9)
24 1nÙ Julia 8 Cantor ±kn¼ê
Ïd, |aj/z| < 1. daq?Ø, XJ i < j ≤ n − 1 z ∈ Ωi, K± |z/aj| < 1.
XJ 1 ≤ j < i, âÚn 3.2.2(1)(2) Ú (3.2.8), ·k
|aj/z|Dj ≤ |ai/z|5|aj/ai|5 < K ε5(i−j)/2 ≤ K−9. (3.2.10)
aq/, XJ i < j ≤ n− 1, ·k
|z/aj|Dj ≤ |z/ai|5|ai/aj|5 < K ε5(j−i)/2 ≤ K−9. (3.2.11)
â½Â, ·k ∑1≤j<i
(−1)n−jDj + (−1)nd1 + (−1)n−idi = 0. (3.2.12)
â (3.2.5), (3.2.10), (3.2.11), (3.2.12) ±9 ε5/2 ≤ K−10, ·k
|Gi(z)| =
∣∣∣∣∣ ∑1≤j<i
(−1)n−jDj
1− (aj/z)Dj+
∑i<j≤n−1
(−1)n−j−1Dj(z/aj)Dj
1− (z/aj)Dj+ (−1)nd1 + (−1)n−idi
∣∣∣∣∣≤ 2K
∣∣∣∣∣ ∑1≤j<i
(−1)n−j(aj/z)Dj
1− (aj/z)Dj+
∑i<j≤n−1
(−1)n−j−1(z/aj)Dj
1− (z/aj)Dj
∣∣∣∣∣<
4K2
1−K−9
n−1∑k=1
ε5k/2 <4K2
1−K−9
ε5/2
1− ε5/2< 5K2 ε5/2,
ùL²XJ z ∈ Ωi, â (3.2.7) ÚÚn 3.2.2(1), ·k
|zDi − aDii | · |Gi(z)|/di+1 < 3K3 ε5/2|ai|Di < ε|ai|Di . (3.2.13)
â (3.2.6) ±9 Rouche ½n, 3§ (3.2.3) ) wi,j ¦éz 1 ≤ j ≤ Di
k wi,j ∈ Ωi. AO/, âÚn 3.2.1(2) 1(Ø |wi,j − wi,j| < ε|ai|. 5¿é1 ≤ i ≤ n− 2, ·k
|ai+1| − |ai| − 2ε|ai| − 2ε|ai+1| > |ai+1|(1− 2ε− (1 + 2ε)K−2) > 0. (3.2.14)
âÚn 3.2.2(1) Ú ri = Di
√di/di+1 ≤ (K/2)1/5, ·k,
ri|ai| sin(π/Di)
ε|ai|≥ K4(
2
K)1/5 · 2
π· π
2K> K2 > 1. (3.2.15)
ùL² wi1,j1 = wi2,j2 = (i1, j1) = (i2, j2). Úny²..
éu 1 ≤ i ≤ n− 1, - CPi := wi,j : 1 ≤ j ≤ Di Ún 3.2.3 ¥ f Cu±
Tri|ai| Di gd.:8Ü¿P CVi = f(CPi).
§3.2. .: ±9V/ 25
Ún 3.2.4. éz 1 ≤ i ≤ n− 1, 3¹ CPi ∪ Tri|ai| ∪ T|ai| Ai, ¦
(1)XJ p = 1, KéuÛê n− ik f(Ai) ⊂ Ds éuóê n− ikf(Ai) ⊂ C\DK.
AO/, f .¤8Ü÷v⋃n−1i=1 CVi ⊂ Ds ∪C \DK. d, Ds C \DK
á3 f Fatou 8¥¿ f−1(As,K) ⊂ As,K.
(2)XJ p = 0, Kéuóê n−ik f(Ai) ⊂ Ds éuÛê n−ikf(Ai) ⊂ C\DM ,
Ù¥ M = (2/s)1/dn. AO/, f .¤8Ü÷v⋃n−1i=1 CVi ⊂ Ds ∪ C \ DM . d
, Ds C \ DM á3 f Fatou 8¥¿ f−1(As,M) ⊂ As,M .
y². - ε = u2K ≤ K−4 Ñy3Ún 3.2.3 ¥ê. éz 1 ≤ i ≤ n− 1, ½Â
Ai = z : (minri, 1 − 2ε)|ai| < |z| < (maxri, 1+ 2ε)|ai| (3.2.16)
Ù¥ ri = Di
√di/di+1. w,, Ai ⊃ CPi ∪ Tri|ai| ∪ T|ai|. â½Â, ·k
(2/K)1Di ≤ minri, 1 ≤ maxri, 1 ≤ (K/2)
1Di . (3.2.17)
XJ z ∈ Ai, ·k
|ai/z| ≤1
(2/K)1Di − 2ε
≤ (K/2)1Di
1− 2K−4(K/2)1/5< (K/2)
1Di (1 + 4/K19/5). (3.2.18)
|z/ai| ≤ (K/2)1Di + 2ε ≤ (K/2)
1Di + 2/K4 < (K/2)
1Di (1 + 1/K3). (3.2.19)
ù¿X
|ai/z|5 < (K/2)5Di (1 + 4/K19/5)5 < (K/2) e20/K19/5
< (K/2) e20/319/5 < 7K/10. (3.2.20)
¿Ó,
|z/ai|5 < (K/2)5Di (1 + 1/K3)5 < (K/2) e5/K3
< (K/2) e5/27 < 7K/10. (3.2.21)
d, aqu (3.2.20) Ú (3.2.21) ?Ø, ·k
|ai/z|di + |z/ai|di+1 < 7K/5. (3.2.22)
5¿éz 1 ≤ i ≤ n−2k |ai/ai+1|di+1 = u |an−1|dn = v. - 1 ≤ i1 ≤ i2 ≤ n−1
26 1nÙ Julia 8 Cantor ±kn¼ê
Ú p ∈ 0, 1, ·k
i2∏j=i1
|aj|(−1)n−j−pDj = |ai1|(−1)n−i1−pdi1 |ai2|(−1)n−i2−pdi2+1
i2−1∏j=i1
∣∣∣∣ ajaj+1
∣∣∣∣(−1)n−j−pdj+1
= |ai1|(−1)n−i1−pdi1 |ai2|(−1)n−i2−pdi2+1 u(−1)n−i1−p−(−1)n−i2−p
2
=
(|a1|d1u/v)(−1)p if i1 = 1 and i2 = n− 1 is even
(|a1|−d1/v)(−1)p if i1 = 1 and i2 = n− 1 is odd.
(3.2.23)
â (3.1.2) ±9 (3.2.23) 1ª, ·k
|f(z)| = |zDi − aDii |(−1)n−i−p |z|(−1)n−pd1
i−1∏j=1
|z|(−1)n−j−pDj
n−1∏j=i+1
|aj|(−1)n−j−pDj ·Qi(z)
= |1− (z/ai)Di |(−1)n−i−p |z/ai|(−1)n−i−p+1di |an−1|(−1)1−pdn u
(−1)n−i−p−(−1)1−p2 ·Qi(z)
= v(−1)1−p u(−1)n−i−p−(−1)1−p
2 |(ai/z)di − (z/ai)di+1|(−1)n−i−p ·Qi(z) ≤ v(−1)1−p u
1−(−1)1−p2 (|ai/z|di + |z/ai|di+1)Qi(z) XJ n− i− p óê
≥ v(−1)1−p u−1−(−1)1−p
2 (|ai/z|di + |z/ai|di+1)−1Qi(z) XJ n− i− p Ûê,
(3.2.24)
Ù¥
Qi(z) =i−1∏j=1
∣∣1− (aj/z)Dj∣∣(−1)n−j−p
n−1∏j=i+1
∣∣1− (z/aj)Dj∣∣(−1)n−j−p
. (3.2.25)
é 1 ≤ i ≤ n− 1, Ä z ∈ Ai. XJ 1 ≤ j < i, â (3.2.20), ·k
|aj/z|Dj ≤ |ai/z|5|aj/ai|5 < 7K ε5(i−j)/2/10 < K−9. (3.2.26)
XJ i < j ≤ n− 1, Kd (3.2.21)
|z/aj|Dj ≤ |z/ai|5|ai/aj|5 < 7K ε5(i−j)/2/10 < K−9. (3.2.27)
du 0 < x ≤ 1 k ex < 1 + 2x ε ≤ K−4, â (3.2.25)–(3.2.27), ·k
Qi(z) <∞∏k=1
(1 + 7K ε5k/2/5
)2 ≤ exp
(14K ε5/2/5
1− ε5/2
)< 1 +K−5 < 1.01. (3.2.28)
Qi(z) >∞∏k=1
(1 + 7K ε5k/2/5
)−2> 1/1.01 > 0.99. (3.2.29)
§3.2. .: ±9V/ 27
éu p = 1, â 3.2.2(2)(3a), éz 1 ≤ i ≤ n− 1, XJ |z| ≤ s, ·k
|zDi/aDii | ≤ |s/a1|Di |a1/ai|Di ≤ (sK−3/2)5K u
5(i−1)K . (3.2.30)
XJ·5¿Ún 3.2.2(1), K
n−1∑i=1
|zDi/aDii | ≤(sK−3/2)
5K
1− u 5K
≤ K10K−10
1−K−10< 1/200. (3.2.31)
XJ p = 0, âÚn 3.2.2(2)(4b), éz 1 ≤ i ≤ n− 1, |z| ≤ s , ·k
|zDi/aDii | ≤ |s/a1|Di |a1/ai|Di ≤ (u1/2/2)5K u
5(i−1)K . (3.2.32)
âÚn 3.2.2(1), K
n−1∑i=1
|zDi/aDii | ≤(u1/2/2)
5K
1− u 5K
≤ K−5
1−K−10< 1/200. (3.2.33)
du 0 ≤ |a| ≤ 1/2 , (1 + 2|a|)−1 ≤ |1 + a|±1 ≤ 1 + 2|a|. â (3.2.31) Ú
(3.2.33), ·
n−1∏i=1
∣∣1− zDi/aDii ∣∣(−1)n−i−p ≤n−1∏i=1
(1 + 2|z/ai|Di
)< e1/100 < K. (3.2.34)
Ïd,
n−1∏i=1
∣∣1− zDi/aDii ∣∣(−1)n−i−p ≥n−1∏i=1
(1 + 2|z/ai|Di
)−1> e−1/100 > 1/K. (3.2.35)
(1) ·ÄkĹ p = 1. XJ n− i ´Ûê, â (3.2.22), (3.2.24) Ú (3.2.28),
z ∈ Ai ·k|f(z)| ≤ v · (7K/5) · 1.01 < 2Kv < s. (3.2.36)
XJ n− i ´óê, â (3.2.22), (3.2.24) Ú (3.2.29), z ∈ Ai ·k
|f(z)| ≥ (v/u) · (7K/5)−1 · 0.99 > v/(2Ku) > K. (3.2.37)
XJ n ´Ûê, âÚn 3.2.2(3a), (3.2.23) Ú (3.2.34), éz z ¦ |z| ≤ s, ·
k
|f(z)| = |z|d1n−1∏i=1
|ai|Di(−1)n−i−1n−1∏i=1
∣∣∣∣1− zDi
aDii
∣∣∣∣(−1)n−i−1
< |s/a1|d1vu−1 · 1.02 < s.
28 1nÙ Julia 8 Cantor ±kn¼ê
ùL²XJ n ´Ûê, K f(Ds) ⊂ Ds. XJ n ´óê |z| ≤ s, âÚn 3.2.2(3b),
(3.2.23) Ú (3.2.35), ·k
|f(z)| = |a1/z|d1vn−1∏i=1
∣∣∣∣1− zDi
aDii
∣∣∣∣(−1)n−i−1
> |a1/s|d1v/1.02 > K.
Ïdéu n óê, f(Ds) ⊂ C \ DK .
5¿ f 3 DK ¡ “~C” z 7→ zdn ,ù´Ï |ai|Di 'u@ |z| ≥ K
z ó~, Ù¥ 1 ≤ i ≤ n − 1. ¯¢þ, daqu (3.2.34)–(3.2.35) ?Ø,
XJ |z| ≥ K, K
|f(z)| ≥ |z|dnn−1∏i=1
(1 + 2
|ai|Di|z|Di
)−1
> K. (3.2.38)
ùL² f(C \ DK) ⊂ C \ DK . l éz n ≥ 2, k f−1(As,K) ⊂ As,K (ã 3.1).
(2) y3·Ä¹ p = 0. XJ n− i ´óê, â (3.2.22), (3.2.24), (3.2.28) Ú
Ún 3.2.2(4a), XJ z ∈ Ai ·k
|f(z)| ≤ v−1u · (7K/5) · 1.01 < 2Ku/v < s. (3.2.39)
XJ n− i ´Ûê, â (3.2.22), (3.2.24), (3.2.29) ÚÚn 3.2.2(4a), éu z ∈ Ai ·k
|f(z)| ≥ v−1 · (7K/5)−1 · 0.99 > 1/(2Kv) > M, (3.2.40)
Ù¥ M = (2/s)1/dn .
XJ n ´óê, âÚn 3.2.2(4b), (3.2.23) Ú (3.2.34), éz¦ |z| ≤ s z, ·
k
|f(z)| = |z|d1n−1∏i=1
|ai|Di(−1)n−in−1∏i=1
∣∣∣∣1− zDi
aDii
∣∣∣∣(−1)n−i
< |s/a1|d1v−1 · e1/100 < s.
ùL²XJ n óê, K f(Ds) ⊂ Ds. XJ n Ûê |z| ≤ s, âÚn 3.2.2(4c),
(3.2.23) Ú (3.2.35), ·k
|f(z)| = |a1/z|d1uv−1
n−1∏i=1
∣∣∣∣1− zDi
aDii
∣∣∣∣(−1)n−i
≥ |a1/s|d1uv−1 · e−1/100 > M.
ÏdXJ n Ûê, K f(Ds) ⊂ C \ DM .
XJ |z| ≥M , K
|f(z)| = |z|−dnn−1∏i=1
∣∣∣∣1− aDiizDi
∣∣∣∣(−1)n−i
≤M−dnn−1∏i=1
(1 +
2|ai|Di|z|Di
)< 2M−dn = s. (3.2.41)
ùL² f(C \ DM) ⊂ Ds. l éz n ≥ 2, ·k f−1(As,M) ⊂ As,M .
§3.2. .: ±9V/ 29
0
0 ∞
∞
0∞
0∞
< s s |a1| |an−1|1
K> K
< s s |a1| |an/2| |an−1|1
K> K
VnV1
Un−1U1
VnV1
Un−1U1
ã 3.1 f1,d1,··· ,dn N'X«¿ã,lm©OéAX n´ÛêÚóê
/. Ê(L.:Ú., ã¡.àêâLCqI.
½n 3.1.1 y². ·=y²¹ p = 1 ÏâÚn 3.2.4(2) aq?ر
^uy² p = 0. üå, ·Ó^ f L« f1,d1,··· ,dn . - Ui f−1(D) ¹ ai
ëÏ©|. XJ n − i ´Ûê, K D = Ds; XJ n − i ´óê, K D = C \ DK . âÚ
n 3.2.4(1), .:8 CPi ⊂ Ui Ui ´¹ Ai «. d, âÚn
3.2.4(1) k f(Ui) ∩ f(Ui+1) = ∅, Ù¥ 1 ≤ i < n − 2, u´ Ui ∩ Ui+1 = ∅. ùL²éuØÓ i, j k Ui ∩ Uj = ∅. b Ui k mi >.ëÏ©|. Ï Ui ¥Tk Di .:
f : Ui → D ´NÝu Di ©|CX, l â Riemann-Hurwitz úª
χUi = 2 −mi = DiχD − Di = 0, Ù¥ χ L«î.«5ê. ùL² mi = 2. Ïdéz
1 ≤ i ≤ n− 1, Ui ´7X:.
é 1 ≤ i ≤ n−2,- Vi+1 UiÚ Ui+1m«.N´wÑ f : Vi+1 → As,K ´
NÝu di+1 CXN. 5¿ As,K ´ëÏع., f−1(As,K)
zëÏ©|´. ùL²3ü V1 Ú Vn, ©Oá3 0 Ú U1, Un−1 Ú
∞ m, ¦ f : V1, Vn → As,K ´NÝ©Ou d1 Ú dn CXN. ¯¢þ, f 3
∂U1 Ú ∂Un−1 þ©OäkNÝ d1 Ú dn V1 Ú Vn ¥Ø¹.: (ã 3.1).
¼ê f Julia 8´ J =⋂k≥0 f
−k(As,K). âE, éz 0 ≤ j ≤ n, f−1 :
As,K → Vj z_©|Ñ´/. ùL² J zëÏ©|Ñ´i@3 0 Ú ∞ m;8. Ï J ©|ØU´ü: f ´V, â [59] ¥½n 1.2, J
zëÏ©|Ñ´^ Jordan (¯¢þ´[). ¼ê f 3Ù Julia8©|þÄå
30 1nÙ Julia 8 Cantor ±kn¼ê
XÚÓu n ÎÒ Σn := 0, 1, · · · , n− 1N ü>²£ (one-side shift). AO/, J Ó
u Σn × S1, ù´·¤I Cantor ± (~Xã 1.1). ùÒ¤½n 3.1.1
y².
5 3.2.5. du f ´V, XJ·éëê ai 6Ä, f Julia 8E,´
Cantor ±, Ù¥ 1 ≤ i ≤ n− 1.
½n 3.2.6. éu n ≥ 3, ½n 3.1.1 ¥@ÀJ ai ¦ fp,d1,··· ,dn Julia 8´
Cantor ±, K fp,d1,··· ,dn Ú McMullen N3§éA Julia 8þØUÿÀÝ.
y². Ï fp,d1,··· ,dn 3 Julia©|þÄåXÚÝu nÎÒ Σn := 0, 1, · · · , n−1N ü>²£, McMullen N3Ù Julia ©|þÄåXÚÝu 2 ÎÒ Σ2 :=
0, 1N ü>²£. ùL² n ≥ 3, fp,d1,··· ,dn Ú McMullenN3§éA Julia
8þØUÿÀÝ.
§3.3 Cantor ±. Julia 8mÿÀÝ
ù!¥, ·y²éu?¿½ Julia 8 Cantor ±kn¼ê, ѱ3
(3.1.2) ¥éN fp,d1,··· ,dn ¦ùükn¼ê3§g Julia 8þ´ÿÀ
Ý.
Ún 3.3.1. XJ f ´ Julia 8 Cantor ±kn¼ê. K J(f) þع
.:.
y². b f 3 Julia ©| J0 ¹ f ê d .: c0. K f 3
c0 NCØ´é. ®² f(J0) ´¹ f(c0) Julia ©| [6, Ún
5.7.2]. ÀJ f(c0)ÿÀ U ¦ U ∩ f(J0)´^ü f−1(U)
¹ c0 ©|´ d + 1 é 1 /N÷ U . 5¿ f−1(U ∩ f(J0)) ¹ c0 ©| J ′
´ëϹ3 J0 ¥. , , J ′ äk(G(Ï Ø´^ü. ùb J(f)
´ Cantor ±gñ.
·¡;8 A ⊂ C ©l 0 Ú ∞ XJ 0 Ú ∞ ©Oá3 C \A üØÓ©|¥. - A Ú B üØ;8¿©O©l 0 Ú ∞. XJ A á3 C \B ¹ 0
©|p, KP A ≺ B. - A ©l 0 Ú ∞ , ·^ ∂−A Ú ∂+A L« A
ü>.©|¿¦ ∂−A ≺ ∂+A.
½n 3.3.2. f ´ Julia 8 Cantor ±kn¼ê. K3 p ∈ 0, 1,n ≥ 2 ±9÷v
∑ni=1
1di< 1 ê d1, · · · , dn ¦ f Ú fp,d1,··· ,dn 3§ Julia 8
þ´ÿÀÝ.
§3.3. CANTOR ±. JULIA 8mÿÀÝ 31
y². f Julia 8 J(f) ´ Cantor ±, KâÚn 3.3.1 , f z
±Ï Fatou ©|o´á5o´Ô. ·=[/y²á5 (éA f V)
¹ |^ Cui ó5)ºÔ/ [16].
e5, ·b f ´V. âb f Fatou 8düüëÏ Fatou
©|ÚÙ§©||¤. - D Ú A ©O f üëÏÚëÏ Fatou ©|8
Ü. ·äó f(D) ⊂ D éz A ∈ A Ñ3ê k ≥ 1 ¦ f k(A) ∈ D. (
Ø f(D) ⊂ D ´w,, ù´ÏüëÏ Fatou ©|E,´üëÏ. XJ
f(A1) = A2, Ù¥ A1, A2 ∈ A, Kâ Riemann-Hurwitz úª A1 ع.:. ùL
²z A ∈ AØU´±ÏÏz±ÏáÚ Fatou©|Ì7¹.:.
,¡, â Sullivan ½n, kn¼êvki Fatou . ùäóÒy².
3 Mobius C¿Âe, ·±b 0 Ú ∞ ©Oáu f üüë
Ï Fatou ©| D0 Ú D∞. =, D = D0, D∞. Ï f(D) ⊂ D, Ø5, ·b
f(D0) = D0 Ú f(D∞) = D∞. - f−1(D0) = D0 ∪A1 ∪ · · · ∪Am, Ù¥ A1, · · · , Am m
©l 0 Ú ∞ ¿¦éz 1 ≤ i ≤ m− 1 k Ai ≺ Ai+1. N´wÑ m ≥ 1. Ä
K, D0 ´ØC, K J(f) = ∂D0, ùÚb J(f) ´ Cantor ±gñ.
b deg(f |D0 : D0 → D0) = d1, deg(f |∂−Ai : ∂−Ai → ∂D0) = d2i é 1 ≤ i ≤ m,
k deg(f |∂+Ai : ∂+Ai → ∂D0) = d2i+1. ùL² deg(f) =∑2m+1
j=1 dj. - W1 0u D0
A1 m Wi 0u Ai−1 Ú Ai m«, Ù¥ 2 ≤ i ≤ m. ·k
f(Wi) = C \D0 deg(f |Wi: Wi → C \D0) = d2i−1 + d2i. ùL²3 Fatou
©| Bi ( Wi ¦ f(Bi) = D∞. XJ3 B′i 6= Bi ¦ B′i ( Wi f(B′i) = D∞, K
73 f−1(D0) ©|á3 Wi ¥, ùb A1 ∪ · · · ∪ Am ´ f−1(D0) ¤k
©|¤8Ügñ. Ï TÐ3 Fatou ©| Bi ( Wi ¦ f(Bi) = D∞
deg(f |Bi : Bi → D∞) = d2i−1 + d2i. aq?ر^uy² D∞ ´ f−1(D∞) á
3 C \Am Ã.©|±N÷ D∞ ©|. Ïd, f−1(D∞) = B1 ∪ · · · ∪Bm ∪D∞d deg(f) =
∑2m+1j=1 dj deg(f |D∞) = d2m+1. P C \ (D0 ∪D∞) E. c f−1(E)
d 2m+ 1 ©| E1, · · · , E2m+1 ¤, §÷véu 1 ≤ i ≤ 2m k Ei ≺ Ei+1. N
f : Ei → E ´NÝ di ÃÜCX, Ù¥ 1 ≤ i ≤ 2m+ 1 (ã 3.2).
- n = 2m+1 p = 1. Ïz EiÑ/¹3 E¥mod(Ei) = mod(E)/di,
l â Grotzsch ت∑n
i=1 1/di < 1. e5, ·òE[/N
φ : C→ C ò f Julia 8þÄåXÚÝ f1,d1,··· ,dn Julia 8þ.
üå, ·P f1,d1,··· ,dn F . 5¿ F (0) = 0 Ú F (∞) =∞. 3üüëÏ
Fatou ©| D′0 Ú D′∞, ¦§3 F ^eØC 0 ∈ D′0 Ú ∞ ∈ D′∞. â½n
3.1.1y²,· F−1(D′0) = D′0∪A′1∪· · ·∪A′m,Ù¥ A′1, · · · , A′m m©l 0Ú
∞ ¦éz 1 ≤ i ≤ m− 1 k A′i ≺ A′i+1. d, deg(F |D′0 : D′0 → D′0) = d1,
32 1nÙ Julia 8 Cantor ±kn¼ê
D∞
E2m+1
Am
∞
D0 0
E2m
E2
E3
E1
B1
E B2
A1
D0 0
D∞ ∞
d1 d1
d2
d2
d3
d3
d2m+1
d2m+1
d2m
d2m
W2
W1
ã 3.2 f N'X«¿ã,Ù¥ di, 1 ≤ i ≤ 2m+ 1L« f 3㥤« Fatou
©|>.þNÝ.
deg(F |∂−A′i : ∂−A′i → ∂D′0) = d2i, éu 1 ≤ i ≤ m k deg(F |∂+A′i : ∂+A
′i → ∂D′0) =
d2i+1. - W ′1 0u D′0 ÚA
′1 m« W ′
i 0u A′i−1 Ú A′i m
«, Ù¥ 2 ≤ i ≤ m. KTÐ3 Fatou ©| B′i ( W ′i ¦ F (B′i) = D′∞
deg(F |B′i : B′i → D′∞) = d2i−1 + d2i. u´·k F−1(D′∞) = B′1 ∪ · · · ∪ B′m ∪ D′∞ deg(F |D′∞) = d2m+1. aq/, - E ′ := C \ (D′0 ∪D′∞). 3 F−1(E ′) 2m + 1
©| E ′1, · · · , E ′2m+1 ¦éu 1 ≤ i ≤ 2m k E ′i ≺ E ′i+1. N F : E ′i → E ′ ´N
Ý di CX, Ù¥ 1 ≤ i ≤ 2m+ 1.
â[/Ãâ, ± ∂D0, ∂D∞, ∂D′0, ∂D
′∞ Ú§cÑ´[ û
k½~ê. ùL²3[/N φ0 : C → C ¦ φ0(D0) = D′0
φ0(D∞) = D′∞, Ï φ0(∂D0) = ∂D′0 φ0(∂D∞) = ∂D′∞. d, À φ0 ¦3
∂D0 ∪ ∂D∞ þk φ0 f = F φ0.
y3·E φ0 : E → E ′ J, φE1 : E1 → E ′1 Xe. éz z ∈ E1 \ ∂−E1,
·ÀJü γ : [0, 1] → E ¦ γ(1) = f(z) γ(0) = w ∈ ∂−E. Ï
f : E1 → E ´CXN, 3 γ J, γ : [0, 1] → E1 ¦ γ(1) = z
w := γ(0) ∈ ∂−E1. aq/,Ï F : E ′1 → E ′´CXN,3 φ0(γ) : [0, 1]→ E ′
J, α : [0, 1]→ E ′1¦ α(0) = φ0(w),ù´Ï3 ∂D0 = ∂−E1þ, φ0f = Fφ0.
½Â φE1(z) := α(1). 5¿ f, F Ñ´NÝ d1 XCXN φ0 : E → E ′ ´[
/, u´3 E1 þk φ0 f = F φE1 φE1 : E1 → E ′1 ´[/. y3·é
φ1 : C→ C 3/®²k½Â: φ1|D0= φ0|D0
, φ1|D∞ = φ0|D∞ φ1|E1 = φE1 . ,
, 3 ∂E1 þk φ1 f = F φ1. aq/, 3[/N φE2m+1 : E2m+1 → E ′2m+1,
§3.3. CANTOR ±. JULIA 8mÿÀÝ 33
φ0 : E → E ′ J,¦3 E2m+1 þk φ0 f = F φE2m+1 . ½Â φ1|E2m+1 = φE2m+1 .
K, 3 ∂E2m+1 þk φ1 f = F φ1.
ØÓu E1 Ú E2m+1 ¹, éu 2 ≤ i ≤ 2m, φ0 : E → E ′ J, φEi : Ei → E ′i
3´Ø. ·Äk`² φEi 35. Ø5, b i ´óê. Ï f :
∂−Ei → ∂D∞ Ú F : ∂−E′i → ∂D′∞ Ñ´NÝ di CXN, 3 φ0 : ∂D∞ → ∂D′∞
J, (Ø) φEi : ∂−Ei → ∂−E′i ¦3 ∂−Ei þk φ0 f = F φEi . |^½
 φE1 Ó, 3l Ei E ′i φ0 : E → E ′ J,, ·ÓP¤ φEi ,
¦3 Ei þk φ0 f = F φEi . 5¿ φEi : Ei → E ′i ´[/. ½Â φ1|Ei = φEi .
K, 3⋃2m+1i=1 Ei k φ0 f = F φ1 3
⋃2m+1i=1 ∂Ei þk φ1 f = F φ1.
ÚÎÒ, é 1 ≤ i ≤ m, - D2i−1 := Bi Ú D2i := Ai. Ké 1 ≤ i < j ≤ 2m, ·
k Di ≺ Dj. ·I3⋃2mi=1Di þ½Â φ1. éz Di,Ù¥ 1 ≤ i ≤ 2m,§ü>.
©| ∂+Ei Ú ∂−Ei+1 Ñ´[. Ï φEi Ú φEi+1Ñ´[/N, N φ1|∂+Ei∪∂−Ei+1
k¦ φDi(Di) = D′i [/*Ü φDi : Di → D′i. y3·[/N
φ1 : C→ C, §½Â φ1|Ei := φEi , φ1|Dj = φDj φ1|D0∪D∞ = φ0, Ù¥ 1 ≤ i ≤ 2m+ 1
1 ≤ j ≤ 2m.
e5, ·½Â φ2. Äk, é j ∈ 0, 1, · · · , 2m,∞, - φ2|Dj = φ1. ,·±·
ªJ, φ1 : E → E ′ φ2 : Ei → E ′i, Ù¥ 1 ≤ i ≤ 2m+ 1. , ·u
N φ2 : C→ C ëY5. e¡´[`². y φ2 3 D0 ∪E1 þëY5, ·
Ik φ2|∂−E1 = φ1. u´é φ1 : E → E ′ 5`, =3«J,ª φ2 : E1 → E ′1.
5¿ φ2|D1 = φ1, ·Iu φ2 3>. ∂+E1 þëY5. ¯¢þ, φ0|E Ú φ1|E *dÓÔ φ1|∂E = φ0|∂E, ùL² φ2|∂+E1 = φ1|∂+E1 , ù´Ï φ2|∂−E1 = φ1|∂−E1 . ù`²
φ2 3 ∂+E1 þ´ëY. aq/, é 2 ≤ i ≤ 2m+ 1, ·±J, φ1 : E → E ′ 5
φ2 : Ei → E ′i y φ2 ëY5. nþ, N φ2 : C → C ÷v (1) φ2 ´[/Ù
û K(φ2) = K(φ1); (2) φ2|f−1(D0∪D∞) = φ1; (3) 3⋃2m+1i=1 Ei þk φ1 f = F φ2
Ïd3 f−2(∂D0 ∪ ∂D∞) þk φ2 f = F φ2.
bé, k ≥ 1 ·®² φk, K φk+1 daqu φ1 φ2
½Â. 8B/, ·±[/N φkk≥0 ¦ (1) éu k ≥ 1 k K(φk) =
K(φ1) ≥ K(φ0); (2) é z ∈ f−k(D0 ∪D∞) k φk+1(z) = φk(z); (3) 3 f−k(∂D0 ∪ ∂D∞)
þk φk f = F φk. ù¿X φkk≥0 ´5x. À φkk≥0 Âñ
f, Ù4·P φ∞, K φ∞ ´[/N3⋃k≥0 f
−k(∂D0 ∪ ∂D∞) þk
φ∞ f = F φ∞. d, K(φ∞) ≤ K(φ1). Ï φ∞ ´ëY, φ∞ f = F φ∞ 3⋃k≥0 f
−k(∂D0 ∪ ∂D∞) 4þ¤á, ù4Ò´ f Julia 8. Ïd φ = φ∞ ´·
é[/N, §ò f Julia 8þÄåXÚÝ F þ. ùÒ(å¹
f(D0) = D0 Ú f(D∞) = D∞ y².
34 1nÙ Julia 8 Cantor ±kn¼ê
Ù§n«¹: (1) f(D0) = D∞, f(D∞) = D∞; (2) f(D0) = D∞, f(D∞) = D0; Ú
(3) f(D0) = D0, f(D∞) = D0 ±aq/y².
XJ D0 Ú D∞ ¥k½ü©|´Ô, K3 f 6Ä fε ¦ fε ´
V3éA Julia 8þ, fε Ú f ´ÿÀÝ [16]. duâþ¡y² fε
3 (3.1.2) ¥´k “.” éA, ùL² f 3 (3.1.2) ¥k.. ùÒ(å
½n 3.3.2 Ú 3.1.2 y².
â½n 3.3.2 3V¹ey², ·á=kXeíØ.
íØ 3.3.3. XJëê ai ½n 3.1.1 ¥@ÀJ, Ù¥ 1 ≤ i ≤ n− 1, K fp,d1,··· ,dn
z Julia ©|Ñ´[.
§3.4 Julia 8þÿÀd Cantor ±kn¼êÝa
ù!8´ONÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓ
ÿÀÝaê.
- RatCCd ¤kNÝu d ≥ 2 Julia 8 Cantor ±kn¼ê|¤8
Ü. â½n 3.1.2, ·éuzkn¼ê f ∈ RatCCd , 3 (n + 1)-|
(p, d1, · · · , dn) Ú f éA, Ù¥ p ∈ 0, 1, n ≥ 2,∑n
i=1 di = d ∑n
i=11di< 1. u´·
ke¡íØ.
íØ 3.4.1. XJ d ≤ 4, K RatCCd 8.
y². XJ d ≤ 4, K§|∑n
i=1 di = d∑ni=1
1di< 1
n ≥ 2, di ≥ 2 1 ≤ i ≤ n
Ã).
XJ f ∈ RatCCd , Ù¥ d ≥ 5, ½Â f I8
Index(f) := (p, d1, · · · , dn) : f Ú fp,d1,··· ,dn 3§ Julia 8þÿÀÝ.
8Ü Index(f) ¥¡*dd. ½kn¼ê f ∈ RatCCd ±kõ
I, ù´du f Ú§ “.” fp,d1,··· ,dn üëÏ Fatou ©|UkõéA. ·^
]A 5L«k8 A ¥ê.
§3.4. JULIA 8þÿÀd CANTOR ±kn¼êÝa 35
½n 3.4.2. NÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓÿÀ
Ýaê
N(d) =∑n≥2
](d1, · · · , dn) :n∑i=1
di = d n∑i=1
1
di< 1
+∑n≥3
](d1, · · · , dn) :n∑i=1
di = d,
n∑i=1
1
di< 1, (d1, · · · , dn) = (dn, · · · , d1) n Ûê .
y². é (3.1.2) ¥z, ®² 0 Ú ∞ áu fp,d1,··· ,dn Fatou 8. - D0
Ú D∞ ©O¹ 0 Ú ∞ üëÏ Fatou ©|. ùpk 4 «¹:
(1) XJ p = 1 n Ûê, K f(D0) = D0 f(D∞) = D∞;
(2) XJ p = 1 n óê, K f(D0) = D∞ f(D∞) = D∞;
(3) XJ p = 0 n Ûê, K f(D0) = D∞ f(D∞) = D0;
(4) XJ p = 0 n óê, K f(D0) = D0 f(D∞) = D0.
·òdþ¡ 4«¹íÑ N(d)Oúª. éu (1)Ú (3) nÛê¹,
(p, d1, · · · , dn) du (p, dn, · · · , d1) (p, d1, · · · , dn) Ødu (1− p, d1, · · · , dn), Ù¥
p ∈ 0, 1. éu (2) Ú (4) n óê¹, (p, d1, · · · , dn) du (1− p, dn, · · · , d1)
XJ (d1, · · · , dn) 6= (d′1, · · · , d′n), Kz (p, d1, · · · , dn) Ødu (p, d′1, · · · , d′n), Ù
¥ p ∈ 0, 1. w,, XJ n + m Ûê, (p, d1, · · · , dn) Ødu (p′, d′1, · · · , d′m), Ù¥
p, p′ ∈ 0, 1.éz½ d ≥ 5 p ∈ 0, 1, ½Â
Np1 = (p, d1, · · · , dn) :
n∑i=1
di = d,n∑i=1
1
di< 1, (d1, · · · , dn) = (dn, · · · , d1) n Ûê,
Np2 = (p, d1, · · · , dn) :
n∑i=1
di = d,
n∑i=1
1
di< 1, (d1, · · · , dn) 6= (dn, · · · , d1) n Ûê,
Np3 = (p, d1, · · · , dn) :
n∑i=1
di = d,n∑i=1
1
di< 1 n óê.
5¿ ]N01 = ]N1
1 ]N02 = ]N1
2 , l NÝ d ≥ 5 Cantor ±kn¼ê Julia 8
þØÓÿÀÝaê
N(d) = (]N11 + ]N1
2/2) + (]N01 + ]N0
2/2) + ]N03 = (]N0
1 + ]N02 + ]N0
3 ) + ]N01 ,
ù½n 3.4.2 Òy².
éuz½ d ≥ 5, â½n 3.4.2 ±ÏLqÞéN´/OÑ N(d).
e!L 3.1 Ñ 5 ≤ d ≤ 36 N(d). d, ·ò3e!Ñ N(d)
þe.O.
36 1nÙ Julia 8 Cantor ±kn¼ê
§3.5 Cantor ±kn¼ê Julia 8þÿÀÝaêO
éz n ≥ 1,- P n ⊂ Rnäk n+1º: (0, 0, · · · , 0), (1, 0, · · · , 0), (0, 1, · · · , 0)
Ú (0, 0, · · · , 1) ü I. =,
P n = (x1, · · · , xn) :n∑i=1
xi < 1 xi > 0, Ù¥ 1 ≤ i ≤ n. (3.5.1)
®² P n î¼NÈ´ Vol(P n) = 1/n!. - a ∈ R l > 0. ½Â#I
P na,l = (a+ lx1, · · · , a+ lxn) : (x1, · · · , xn) ∈ P n. (3.5.2)
N´wÑ P na,l NÈ
Vol(P na,l) = ln/n!. (3.5.3)
e¡Ún±â²þت.
Ún 3.5.1. (1) - x, y, α, β 4 ¢ê, K
1
x+
1
y≥ (α + β)2
α2x+ β2y. (3.5.4)
d, Ò¤á= αx = βy.
(2) - xi, 1 ≤ i ≤ n n ¢ê, K
n∑i=1
1
xi≥ n2∑n
i=1 xi. (3.5.5)
d, Ò¤á=éz 1 ≤ i, j ≤ n k xi = xj.
½ê d ≥ 5. éz n ≥ 1, ½Â Rn+ f8
Dn = (x1, · · · , xn) ∈ Rn+ :
n∑i=1
1
xi+
1
d−∑n
i=1 xi< 1, xi > 0 d−
n∑i=1
xi > 0. (3.5.6)
Ún 3.5.2. 8Ü Dn = (n+ 1)2 ≥ d.
y². |^Ún 3.5.1, ·k
n∑i=1
1
xi+
1
d−∑n
i=1 xi≥ 1
(∑n
i=1 xi)/n2
+1
d−∑n
i=1 xi≥ (n+ 1)2
d. (3.5.7)
ùL²XJ (n+ 1)2 ≥ d, K Dn = ∅.b (n + 1)2 ≤ d − 1. ª (3.5.7) ¥Ò¤á=éz 1 ≤ i ≤ n k
xi = dn+1
. AO/, k ( dn+1
, · · · , dn+1
) ∈ Dn. Úny..
§3.5. CANTOR ±kn¼ê JULIA 8þÿÀÝaêO 37
e5, 3vkAO`²¹e, ù!¥·o´b
(n+ 1)2 ≤ d− 1. (3.5.8)
Ún 3.5.3. (1) 8Ü Dn ´à.
(2) 8Ü Dn ¹I P nan,bn−an, Ù¥
an =d+ n2 − 1−
√∆
2n, bn =
d− n2 + 1 +√
∆
2(3.5.9)
∆ = (d− n2 − 1)2 − 4n2.
y². (1) - (x1, · · · , xn), (y1, · · · , yn) ∈ Dn t ∈ [0, 1], Ï x 7→ 1/x 3 (0,∞) þ
´à, ·k
n∑i=1
1
txi + (1− t)yi+
1
d−∑n
i=1(txi + (1− t)yi)
=n∑i=1
1
txi + (1− t)yi+
1
t(d−∑n
i=1 xi) + (1− t)(d−∑n
i=1 yi)
<n∑i=1
t
xi+
n∑i=1
1− tyi
+t
d−∑n
i=1 xi+
1− td−
∑ni=1 yi
< 1.
ùL² Dn ´à.
(2) 5¿e¡ n+ 1 :
(an, an, · · · , an), (bn, an, · · · , an), (an, bn, · · · , an), · · · , (an, an, · · · , bn)
Ñá3 Dn >.þ. Ï Dn ´à,ùL² Dn ¹I P nan,bn−an . ã 3.3`² n = 2
¹.
- (x1, · · · , xn) ∈ Rn. XJz xi Ñ´ê, K¡ù:´ê:, Ù¥
1 ≤ i ≤ n.
Ún 3.5.4. - a ∈ R l ≥ n. K P na,l ¥ê:ê Num(P n
a,l) ÷v
Num(P na,l) ≥
(l − n)n
n!. (3.5.10)
y². y²©¤ a ∈ Z Ú a 6∈ Z ü«¹?Ø. ·3dÑ[!.
38 1nÙ Julia 8 Cantor ±kn¼ê
O an nan bnx1
xn
DnPnan,bn−an
x1 = · · · = xn
ã 3.3 « Dn 3 Rn+ ¥ , Ù¥ n = 2.
íØ 3.5.5. bn − an ≥ n , Dn ¥ê:ê Num(Dn) ÷v
Num(Dn) ≥ (bn − an − n)n
n!. (3.5.11)
Ù¥^ bn − an ≥ n du
1 + 3n+ (1 + n)√
1 + 4n2
2≤ d. (3.5.12)
y². ª (3.5.11) Ún 3.5.4 íØ. ·=y² (3.5.12). â (3.5.9), bn −an ≥ n du
d− n2 + 1 +√
∆
2− d+ n2 − 1−
√∆
2n− n ≥ 0. (3.5.13)
O,
d ≥ 1 + 3n+ (1 + n)√
1 + 4n2
2½ö d ≤ 1 + 3n− (1 + n)
√1 + 4n2
2. (3.5.14)
5¿â (3.5.8) k d ≥ (n+ 1)2 + 1. ùL² (3.5.12) ¤á.
§ ∑n+1
i=1 di = d∑n+1i=1
1di< 1
n ≥ 1, di ≥ 2 1 ≤ i ≤ n+ 1
ê) (d1, · · · , dn, dn+1) ê S(n+ 1) u Dn ¥ê:ê Num(Dn).
- [a] ¢ê a êÜ©. âíØ 3.5.5, ·k
§3.5. CANTOR ±kn¼ê JULIA 8þÿÀÝaêO 39
½n 3.5.6.∑n≥1
S(n+ 1) ≥n0∑n=1
(bn − an − n)n
n!≥
n0∑n=1
(d− n2 − 3n− 1)n
n!, (3.5.15)
Ù¥ n0 = [(√
4d+ 5− 3)/2].
y². â (3.5.9), ·k
√∆ =
√(d− n2 − 1)2 − 4n2 ≥ d− (n+ 1)2.
Ïd, ·k
bn − an − n =d(n− 1) + n+ 1− 3n2 − n3 + (n+ 1)
√∆
2n
≥ d− n2 − 3n− 1.
(3.5.16)
5¿ n ≥ 1 ,
1 + 3n+ (1 + n)√
1 + 4n2
2< n2 + 3n+ 1 (n+ 1)2 < n2 + 3n+ 1. (3.5.17)
Ïd, n ≤ (√
4d+ 5 − 3)/2 , bn − an − n ≥ 0. ª (3.5.15) >ت´díØ
3.5.12 Ñ(Ø.
â (3.5.16), ·k
n0∑n=1
(bn − an − n)n
n!≥
n0∑n=1
(d− n2 − 3n− 1)n
n!,
Ù¥ n0 = [(√
4d+ 5− 3)/2].
5 3.5.7. XJ·Ä¤k¦∑n+1
i=1 di = d Ñ^∑n+1
i=1 1/di < 1
Uê| (d1, · · · , dn, dn+1), Ù¥ di ≥ 2, N´wÑ∑n≥1
S(n+ 1) ≤n1∑n=1
C nd−n−2, (3.5.18)
Ù¥ n1 = [√d− 1]− 1 C n
m = m!/(n!(m− n)!) |Üê.
½n 3.5.8. NÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓÿÀ
Ýaê N(d) ÷v
n0∑n=1
(d− n2 − 3n− 1)n
n!≤ N(d) < 2
n1∑n=1
C nd−n−2, (3.5.19)
Ù¥ n0 = [(√
4d+ 5− 3)/2] n1 = [√d− 1]− 1.
40 1nÙ Julia 8 Cantor ±kn¼ê
y². â½n 3.5.6, 5 3.5.7 Ú½n 3.4.2 ¥ã N(d) úª½n 3.5.8
¤á.
¯¢þ, â½n 3.5.6 Ú½n 3.4.2, N(d) e.Úþ.±de¡°(ê
O:
n0∑n=1
(bn − an − n)n
n!Ú
∑1 ≤ n ≤ n1
n is odd
C nd−n−2 + 2
∑1 ≤ n ≤ n1
n is even
C nd−n−2. (3.5.20)
·Ñe¡L5' 5 ≤ d ≤ 36 , N(d) °(Úd (3.5.20) ª¤(
½ N(d) þe..
d 5 6 7 8 9 10 11 12
LB(d) 2 3 4 5 6 7 10 14
N(d) 2 3 4 5 6 11 22 37
UB(d) 2 3 4 5 6 37 50 65
d 13 14 15 16 17 18 19 20
LB(d) 19 25 33 41 50 61 75 93
N(d) 46 57 68 81 110 159 228 290
UB(d) 82 101 122 145 390 483 590 712
d 21 22 23 24 25 26 27 28
LB(d) 117 147 186 233 290 358 438 531
N(d) 410 519 716 872 1070 1323 1722 2258
UB(d) 850 1005 1178 1370 1582 11505 14040 16978
d 29 30 31 32 33 34 35 36
LB(d) 642 774 935 1130 1370 1662 2018 2450
N(d) 3066 4227 5566 6950 8604 10483 12916 15838
UB(d) 20360 24229 28630 33610 39218 45505 52524 60330
L 3.1 NÝ d ≥ 5 Cantor ±kn¼ê Julia 8þØÓÿÀ
Ýaê N(d) ±9d (3.5.20) ¤ N(d) þe. UB(d) Ú LB(d), Ù¥
5 ≤ d ≤ 36.
§3.6. JULIA 8 CANTOR ±Vkn¼ê 41
§3.6 Julia 8 Cantor ±Vkn¼ê
m,n ≥ 2 ü÷v 1/m+ 1/n < 1 ê, -
Pλ(z) =1n((1 + z)n − 1) + λm+nzm+n
1− λm+nzm+n, (3.6.1)
Ù¥ λ ∈ C∗ = C \ 0. λ ¿©, Pλ ±w¤´e¡Ôõª6Ä:
P (z) =(1 + z)n − 1
n. (3.6.2)
5¿ P 3:k¦fu 1 ÔØÄ: −1 ´Ùêu n − 1 .:.
ùL² P küëÏ Fatou ©|, Ù¥¤k:Ñ 0 ¤áÚ. AO/, P
Julia 8´äkáõk: Jordan .
λ éÿ, ·F"þ¡ã P ,5é Pλ Ó¤á. éw,,
Pλ Ú P m´3éõÉ. ¼ê Pλ NÝ´ m + n Pλ(∞) = −1. §k
2(m+ n)− 2 .:: m− 1 3 ∞, n− 1 ~C −1 e m+ n .:á
3 Tr0/|λ| NC, Ù¥ r0 = m+n√n/m (Ún 3.6.3). ¯¢þ, 3ù!"·ò¬w
Pλ ±w¤´ “Ô” McMullen N, ù´Ï3éA Julia 8þ Pλ Ý
u McMullen N fλ.
Äk, ·y²éz n ≥ 2, P ØÄÔ Fatou ©|¹î¼ D(−34, 3
4)
XJ λ ¿©, K Pλ ò D(−34, 3
4) N\gC.
Ún 3.6.1. (1) éz n ≥ 2, P (D(−34, 3
4)) ⊂ D(−3
4, 3
4) ∪ 0.
(2) XJ 0 < |λ| < 1/(3n), K Pλ(D(−34, 3
4)) ⊂ D(−3
4, 3
4) ∪ 0. AO/, D(−3
4, 3
4) á
3 Pλ ¹ØÄ: 0 Ô Fatou ©|¥.
y². XJ z ∈ D(−34, 3
4), K |P (z) + 1/n| = |1 + z|n/n ≤ 1/n. AO/, ØÒ±
UÒ= z = 0. ùÒy² (1).
(2) y²ò©¤ü«¹: |z| é±9 |z| Ø´é. éz z = −34
+ 34eiθ ∈
∂D(−34, 3
4), Ù¥ −π < θ ≤ π, â (1), ·k |1 + P (z)| ≤ 5/2 |λz|m+n < 1/2, ù´
Ï |λ| < 1/(3n). ùL²
|Pλ(z)− P (z)| =
∣∣∣∣∣λm+nzm+n(1 + P (z))
1− λm+nzm+n
∣∣∣∣∣ ≤ 5|λz|m+n. (3.6.3)
Ï |z| = 34|1− eiθ| = 3
4|e−iθ/2 − eiθ/2| = 3
2| sin θ
2| ≤ 3
4|θ| |λ| < 1/(3n), ·k
|Pλ(z)− P (z)| ≤ 5 (|θ|/(4n))m+n. (3.6.4)
42 1nÙ Julia 8 Cantor ±kn¼ê
,¡, ÏXJ |θ| ≤ π2, K | sin θ| ≥ 2
π|θ|, ·k
|P (z) + 3/4| =∣∣∣∣(1
4+ 3
4eiθ)n − 1
n+
3
4
∣∣∣∣ ≤∣∣1
4+ 3
4eiθ∣∣n − 1
n+
3
4
=(1− 3
4sin2 θ
2)n/2 − 1
n+
3
4≤
(1− 3θ2
4π2 )n/2 − 1
n+
3
4.
(3.6.5)
XJ |θ| < 2π/n, K 3θ2
4π2 <2n. âÚn 3.2.1(3), ·k
|P (z) + 3/4| ≤ −n2· 3θ2
4π2
3n+
3
4=
3
4− θ2
8π2. (3.6.6)
Ïd, (Ü (3.6.4) Ú (3.6.6), XJ |θ| < 2π/n, K
|Pλ(z) + 3/4| ≤ |P (z) + 3/4|+ |Pλ(z)− P (z)| ≤ 3
4− θ2
8π2+ 5
(|θ|4n
)m+n
≤ 3/4. (3.6.7)
XJ 2π/n ≤ |θ| ≤ π, â (3.6.5) Ú (3.6.6), ·
|P (z) + 3/4| ≤ 3
4− 1
2n2. (3.6.8)
â (3.6.4) Ú (3.6.8), ±XJ 2π/n ≤ |θ| ≤ π, K
|Pλ(z) + 3/4| ≤ 3
4− 1
2n2+ 5 (
|θ|4n
)m+n < 3/4. (3.6.9)
ÃØN, ·®²y²éz z ∈ ∂D(−34, 3
4)k |Pλ(z) + 3
4| ≤ 3
4 |Pλ(z) + 3
4| = 3
4
= z = 0. y²..
§3.2L§, ·5(½ Pλ gd.: .ÏLO, Pλ k
. m+ 2n− 1 .:´e¡§):
(1 + z)n−1 + λm+nzm+n−1(1 +m/n)[(1 + z)n + n− 1]− z(1 + z)n−1 = 0. (3.6.10)
Ún 3.6.2. XJ 0 < |λ| < 1/(3n), K Pλ k n − 1 .:á3 D(−1, |λ|) (D(−3
4, 3
4) ¥.
y². XJ |z + 1| ≤ |λ| < 13n
, K |z| · |1 + z|n−1 ≤ (1 + |λ|)|λ|n−1 < 1
(1 +m/n) |(1 + z)n + n− 1| ≤ (1 +m/n)(|λ|n + n− 1) < m+ n. (3.6.11)
ùL²XJ |z + 1| ≤ |λ|, K∣∣λm+nzm+n−1(1 +m/n)[(1 + z)n + n− 1]− z(1 + z)n−1∣∣
< |λ|n−1 · |λz|m−1|λ|2|z|n(m+ n+ 1) < |λ|n−1 · (2n)1−m(9n2)−1e1/3(m+ n+ 1)
< |λ|n−1 · (m+ n− 1)/(2n)m+1 < |λ|n−1.
(3.6.12)
â Rouche ½nÚ (3.6.10), ½ny.
§3.6. JULIA 8 CANTOR ±Vkn¼ê 43
- CP := wj = r0λ
exp(πi 2j−1m+n
) : 1 ≤ j ≤ m+ n mλm+nzm+n + n = 0 ":8,
Ù¥ r0 = m+n√n/m. Ï h(x) = x1/x, x > 0 3 x = e ? e1/e < 3/2, ·k
2/3 < 1/ m√m < r0 <
n√n < 3/2. (3.6.13)
e¡ÚnL« Pλ m+ n .:Ñ “~C” CP .
Ún 3.6.3. XJ0 < |λ| < 1/(2mn2), K (3.6.10) k) wj ¦ |wj − wj| <2(m+ n)/m, Ù¥ 1 ≤ j ≤ m+ n. d, wi = wj = i = j.
y². 3 (3.6.10) ü>Óر (1 + z)n−1, ·k
1 + λm+nzm+n−1
(m
nz +
m+ n
n
(1 +
n− 1
(1 + z)n−1
))= 0. (3.6.14)
½ö, ¤k^/ª
n
mλm+n+ zm+n +
(m+ n)zm+n−1
m
(1 +
n− 1
(1 + z)n−1
)= 0. (3.6.15)
- Ω = z : |zm+n + nmλ−(m+n)| ≤ β|λ| · n
m|λ|−(m+n), Ù¥ β = 2(m+n)
mr0< 3(m+n)
m. X
J z ∈ Ω, K |λm+nzm+n + nm| < β|λ| · n
mâÚn 3.2.1(2) |z − wj| < βr0 é,
1 ≤ j ≤ 2n ¤á. XJ z ∈ Ω 0 < |λ| < 1/(2mn2), ·k
n− 1
|1 + z|n−1<
n− 1
((|λ|−1 − β)r0 − 1)n−1<
n− 1
(2m+1n2/3− 3− 2n/m)n−1<
1
15(3.6.16)
β|λ| ≤ 2(m+ n)
2mn2 ·mr0
<3
2mn
(1
m+
1
n
)<
1
4, ¤±
1 + β|λ|2(1− β|λ|)
<5
6. (3.6.17)
Ïd, XJ z ∈ Ω 0 < |λ| < 1/(2mn2), â (3.6.16) Ú (3.6.17), ·k∣∣∣∣(m+ n)zm+n−1
m
(1 +
n− 1
(1 + z)n−1
)∣∣∣∣ =m+ n
m|λ|m+n
∣∣∣∣λm+nzm+n
z
(1 +
n− 1
(1 + z)n−1
)∣∣∣∣<
m+ n
m|λ|m+n
(β|λ|+ 1)n/m
r0(1/|λ| − β)· 16
15=
nβ|λ|m|λ|m+n
1 + β|λ|2(1− β|λ|)
· 16
15<
nβ|λ|m|λ|m+n
.
(3.6.18)
é (3.6.15) A^ Rouche ½n,âÚn 3.2.1(2), Ún1Ü©y²Ò(å.
aqu (3.2.15) ?Ø, XJ 0 < |λ| < 1/(2mn2), ·k
(r0/|λ|) · sin(π/(m+ n))
2(m+ n)/m≥ mr0
(m+ n)2|λ|>
2m+1m
3(m/n+ 1)2> 1. (3.6.19)
ùL² wi = wj = i = j. Úny..
44 1nÙ Julia 8 Cantor ±kn¼ê
- CP := wj : 1 ≤ j ≤ m + n Pλ C± Tr0/|λ| m + n .:P
CV := Pλ(wj) : 1 ≤ j ≤ m+ n. - CP−1 Pλ C −1 n− 1 .: (Ún
3.6.2) P CV−1 = Pλ(z) : z ∈ CP−1.P T0 Pλ ¹:á5s Fatou ©|- U := D(−3
4, 3
4). âÚn
3.6.1(2) Ú 3.6.2, · CP−1 ∪ CV−1 ⊂ U ⊂ T0. du Pλ(∞) = −1, ùL²3 ∞¦ Pλ òùN −1 . - T∞ ¦ ∞ ∈ T∞ Fatou
©| U0, U∞ ©O¦ 0 ∈ U0 ∞ ∈ U∞ P−1λ (U) ëÏ©|. éw,, ·k
U ⊂ U0 ⊂ T0 U∞ ⊂ T∞.
Ún 3.6.4. XJ 0 < |λ| ≤ 1/(210mn3), K3¹ T1/|λ| ∪ CP A1, ¦
Pλ(A1) ⊂ U ′∞ ⊂ U∞, Ù¥ U ′∞ ´ ∞ .
y². âÚn 3.6.3 CP “A” þ!/á3± Tr0/|λ| þ¿ Pλ ¤kk
4:Ñá3± T1/|λ| þ. ½Â
A1 = z : 1/(2|λ|) < |z| < 2/|λ|. (3.6.20)
5¿r0
|λ|+
2(m+ n)
m<
3
2|λ|+ 2 +
2n
m<
2
|λ|(3.6.21)
r0
|λ|− 2(m+ n)
m>
2
3|λ|− 2− 2n
m>
1
2|λ|. (3.6.22)
âÚn 3.6.3, ·k T1/|λ| ∪ CP ⊂ A1. XJ z ∈ A1 |λ| ≤ 1210mn3 , K
|Pλ(z) + 1| ≥ (|z| − 1)n
n(|λz|m+n + 1)≥
( 12|λ| − 1)n
n(2m+n + 1)=
(1− 2|λ|)n
2nn|λ|n(2m+n + 1)>
2
|λ|1+ nm
+ 1.
(3.6.23)
¯¢þ,
(1− 2|λ|)n
2m+n + 1>
(1− 2210mn3 )n
2m+n + 1>
0.9
2m+n + 1>
1
2m+n+1+ 2nn|λ|n. (3.6.24)
Ï |λ| ≤ 1210mn3 , ùL² (3.6.23) ±deª:
2m+2n+2 n |λ|n ≤ |λ|1+n/m. (3.6.25)
y3·®²y²: XJ z ∈ A1 |λ| ≤ 1210mn3 , K |Pλ(z)| > 2
|λ|1+n/m .
,¡, XJ |z| ≥ 2|λ|1+n/m , K
|Pλ(z) + 1| ≤ (|z|+ 1)n + 1
|λz|m+n − 1≤ (1 + |z|−1)n + |z|−n
2m − |z|−n<
1
2. (3.6.26)
§3.6. JULIA 8 CANTOR ±Vkn¼ê 45
U ′∞
T∞
U∞
T0
U0−1
0
∞ ∞U ′∞
T∞
T0
−1
0
U
ã 3.4 Pλ N'X«¿ã. Ê(L«.:.
ùL² Pλ(z) ∈ D(−1, 12) ⊂ U . - U ′∞ P−1
λ (D(−1, 12)) ¹ z : |z| ≥ 2
|λ|1+n/m ©|, ù`² Pλ(A1) ⊂ U ′∞ ⊂ U∞ (ã 3.4).
½n 3.1.3 y². éz¦ 0 < |λ| ≤ 1/(210mn3) λ. - A := C \ (U ∪ U ′∞).
Ï Pλ : U ′∞ → D(−1, 12) ´NÝu m _;N, ùL² U ′∞ ´üëÏ A ´
.Ï P−1λ (U ′∞)¹ m+n.:¿ Pλ 3þ¡NÝ´ m+n,¤±
P−1λ (U ′∞) ´. ùL² P−1
λ (A) düØ I1 Ú I2 ¤ I1 ∪ I2 ⊂ A.
Pλ 3 I1 Ú I2 NÝ©O m Ú n.
e¡?Ø~aqu½n 3.1.1 y². Pλ Julia 8´ Jλ =⋂k≥0 P
−kλ (A).
âE,éu j = 1½ 2, P−1λ : A→ Ij ´/, l Jn ëÏ©|´i@3 −1Ú∞
m;8. du Jn ëÏ©|ØU´ü:, [59] ¥½n 1.2 y²ÓéAÛk
kn¼ê¤á ( [59, §9] Ú [73]), · Jn zëÏ©|Ñ´ Jordan .
¼ê Pλ 3Ù Julia 8©|þÄåXÚÓu 2 ÎÒ Σ2 := 0, 1N ü>²£. A
O/, Jλ Óu Σ2 × S1, ù´ Cantor ±.
5: â½n 3.1.3 y²¿(ܽn 3.3.2, · Pλ 3 Julia 8þÄåXÚ
Ýu, fλ Julia 8þÄåXÚ, Ù¥ fλ (3.1.1) ¥½Â McMullen N. Ï
d, ·±ò Pλ w¤´ “Ô” McMullen N, ÏOÒ´ gη
áÚ9Ùcd Pλ ÔÚAcO (ã §3.6).
46 1nÙ Julia 8 Cantor ±kn¼ê
ã 3.5 Pλ Julia 8, Ù¥ m = 3, n = 2 λ v¦ Jλ ´ Cantor
±. Pλ ¤k Fatou©|ÑSÙ¥äkÔØÄ: 1ØC
Ô Fatou ©|¥ (ã¥mÜ©¤« “ès”).
§3.7 õV Cantor ±. Julia 8~f
ù!, ·òEõVkn¼ê, ¦¦ Julia 8´ Cantor ±.
ɽn 3.1.1 éu, éz n ≥ 2, ·½Â
Pn(z) = An(n+ 1)z(−1)n+1(n+1)
nzn+1 + 1
n−1∏i=1
(z2n+2 − b2n+2i )(−1)i−1
+Bn, (3.7.1)
Ù¥ |bi| = si, 0 < s ≤ 1/(25n2)
An =1
1 + (2n+ 2)Cn
n−1∏i=1
(1−b2n+2i )(−1)i , Bn =
(2n+ 2)Cn1 + (2n+ 2)Cn
Cn =n−1∑i=1
(−1)i−1b2n+2i
1− b2n+2i
.
(3.7.2)
Ún 3.7.1. (1) Pn(1) = 1 P ′n(1) = 1.
(2) 1− s2n+1/(n+ 1) < |An| < 1 + s2n+1/(n+ 1) |Bn| < s2n+1/(3n+ 3).
y². ÏLO± Pn(1) = 1. 5¿
Fn(z) :=zP ′n(z)
Pn(z)−Bn
=n−1∑i=1
(−1)i−1(2n+ 2)z2n+2
z2n+2 − b2n+2i
+ (−1)n+1(n+ 1)− n(n+ 1)zn+1
nzn+1 + 1.
(3.7.3)
§3.7. õV CANTOR ±. JULIA 8~f 47
ùL²
P ′n(1)
Pn(1)−Bn
= (2n+ 2)n−1∑i=1
(−1)i−1b2n+2i
1− b2n+2i
+ (2n+ 2)n−1∑i=1
(−1)i−1 + (−1)n+1(n+ 1)− n
= (2n+ 2)n−1∑i=1
(−1)i−1b2n+2i
1− b2n+2i
+ 1 := (2n+ 2)Cn + 1.
(3.7.4)
Ïd, ·k
P ′n(1) = (1−Bn)((2n+ 2)Cn + 1) = 1. (3.7.5)
ùL² 1 ´ Pn ÔØÄ:. ùÒ¤ (1) y².
éu (2),Ïé 1 ≤ i ≤ n−1k |1−b2n+2i |−1 ≤ 1+2|b1|2n+2 0 < s ≤ 1/(25n2) ≤
1/100, u´
(2n+ 2)|Cn| < (2n+ 2) (1 + 2|b1|2n+2)n−1∑i=1
|bi|2n+2 ≤ (2n+ 2)(1 + 2s2n+2)s2n+2
1− s2n+2<
s2n+1
4n+ 4.
(3.7.6)
·k
|Bn| =∣∣∣∣ (2n+ 2)Cn1 + (2n+ 2)Cn
∣∣∣∣ < (2n+ 2)|Cn|(1 + (4n+ 4)|Cn|) <s2n+1
3n+ 3(3.7.7)
|An| < (1+(4n+4)|Cn|)n−1∏i=1
(1+2|bi|2n+2) < (1+s2n+1
2n+ 2)(1+5s2n+2) < 1+
s2n+1
n+ 1. (3.7.8)
d, ·k
|An| > (1− (2n+ 2)|Cn|)n−1∏i=1
(1− |bi|2n+2) > (1− s2n+1
4n+ 4)(1− s2n+2
1− s2n+2) > 1− s2n+1
n+ 1.
(3.7.9)
Úny..
·Äk)ºeE¡g. éu n ≥ 2, ½Â Q(z) = (zn+1 + n)/(n+ 1)
ϕ(z) = 1/z, K Q(z) := ϕ Q ϕ−1(z) = (n + 1)zn+1/(nzn+1 + 1) ÷v: ∞ ´ Q ê
n .:, T.:ÔØÄ: 1 ¤áÚ. Ï bi1≤i≤n−1 Ñé, kn¼ê
Pn ±w´ Q 6Ä. An Ú Bn y 1 o´ Pn ÔØÄ: (Ú
n 3.7.1). ±y² Pn ò T|bi| N\ T0 ½ö T∞, ùûu i´Ûê
48 1nÙ Julia 8 Cantor ±kn¼ê
´óê, Ù¥ T0 Ú T∞ ©OL«¹ 0 Ú ∞ Fatou ©| (Ún 3.7.5). Fatou ©|
T∞ o´Ô, T0 ´á5½öS T∞ ûu n ´Ûê´óê. ½n 3.1.4
y²òæ^cü!aq.
XJ |z| ≤ 1, K |Q(z)| ≤ 1. ùL²éz n ≥ 2, Q ØÄÔ Fatou ©|¹
ü . Ïd, Q Ô Fatou ©|¹4ü Ü C \D. ¦+õª Q
®²6Ä Pn, ·E,ke¡Ún.
Ún 3.7.2. Pn(C \D) ⊂ (C \D) ∪ 1. AO/, C \D á3 Pn >.þ¹k
ÔØÄ: 1 Ô Fatou ©|p.
Ún 3.7.2 y²'æ, ·3e!.
Ún 3.7.3. é n ≥ 2 Ú 1 ≤ i ≤ n− 1, K
∑1≤j<i
(−1)j +∑
i<j≤n−1
(−1)j−1 +1 + (−1)n+1
2= 0. (3.7.10)
y². ?ØÄuL 3.2 ¥Ñ¹.
∑16j<i(−1)j
∑i<j6n−1(−1)j−1 (1 + (−1)n+1)/2
Ûê n Ûê i 0 −1 1
óê i −1 0 1
óê n Ûê i 0 0 0
óê i −1 1 0
L 3.2 Ún 3.7.3 y².
±c, ·ÄkéÑ Pn .: . 5¿ 0 Ú ∞ Ñ´ Pn ê n
.: Pn NÝ´ n2 + n. Pn 2(n2− 1) .:Ñ´§ Fn(z) = 0
) ( (3.7.3)).
éu 1 ≤ i ≤ n− 1, - CP i := wi,j = bi exp(πi 2j−12n+2
) : 1 ≤ j ≤ 2n+ 2 þ!©Ù3± T|bi| þ 2n+ 2 :8Ü. e¡ÚnaquÚn 3.2.3 Ú 3.6.3.
Ún 3.7.4. éz wi,j ∈ CP i, Ù¥ 1 ≤ i ≤ n − 1 1 ≤ j ≤ 2n + 2, 3
Fn(z) = 0 ) wi,j, ¦ |wi,j − wi,j| < sn+1/2|bi|. d, wi1,j1 = wi2,j2 =
(i1, j1) = (i2, j2).
§3.7. õV CANTOR ±. JULIA 8~f 49
y². 5¿ Fn(z) = 0 du
n−1∑i=1
(−1)i−1 z2n+2 + b2n+2
i
z2n+2 − b2n+2i
+1 + (−1)n+1
2− nzn+1
nzn+1 + 1= 0. (3.7.11)
3 (3.7.11) ü>Ó¦± z2n+2 − b2n+2i , Ù¥ 1 ≤ i ≤ n− 1, ·k
(−1)i−1(z2n+2 + b2n+2i ) + (z2n+2 − b2n+2
i )Gi(z) = 0, (3.7.12)
Ù¥
Gi(z) =∑
1≤j≤n−1, j 6=i
(−1)j−1z2n+2 + b2n+2
j
z2n+2 − b2n+2j
+1 + (−1)n+1
2− nzn+1
nzn+1 + 1. (3.7.13)
- Ωi = z : |z2n+2 + b2n+2i | ≤ sn+1/2|bi|2n+2, Ù¥ 1 ≤ i ≤ n − 1. XJ z ∈ Ωi,
KâÚn 3.2.1(2) k |z|n+1 ≤ (1 + sn+1/2)|bi|n+1 ≤ (1 + sn+1/2)sn+1. Ïd, d s ≤1/(25n2) ≤ 1/100 ∣∣∣∣ nzn+1
nzn+1 + 1
∣∣∣∣ ≤ n(1 + sn+1/2)sn+1
1− n(1 + sn+1/2)sn+1≤ (1 + 100−5/2)sn+1/2/5
1− (1 + 100−5/2)100−5/2/5< 0.3 sn+1/2.
éz z ∈ Ωi, XJ 1 ≤ j < i, ·k
|z/bj|2n+2 = |z/bi|2n+2|bi/bj|2n+2 < (1 + sn+1/2) s(2n+2)(i−j). (3.7.14)
XJ i < j ≤ n− 1, âÚn 3.2.1(2) 1ã, ·k
|bj/z|2n+2 = |bi/z|2n+2|bj/bi|2n+2 ≤ (1 + 2 · sn+1/2) s(2n+2)(j−i). (3.7.15)
â (3.7.14), (3.7.15) ÚÚn 3.7.3, ·k∣∣∣∣Gi(z) +nzn+1
nzn+1 + 1
∣∣∣∣=
∣∣∣∣∣ ∑1≤j<i
(−1)j1 + (z/bj)
2n+2
1− (z/bj)2n+2+
∑i<j≤n−1
(−1)j−1 1 + (bj/z)2n+2
1− (bj/z)2n+2+
1 + (−1)n+1
2
∣∣∣∣∣< 3 · (1 + 2 · sn+1/2)
(∑1≤j<i
s(2n+2)(i−j) +∑
i<j≤n−1
s(2n+2)(j−i)
)< 6 · (1 + 2 · sn+1/2)2 s2n+2.
(3.7.16)
(3.7.16) ¥1ت¤á´ÏXJ x < 1/3 Kk 2x/(1 − x) ≤ 3x (ùp x ≤(1 + 2 · sn+1/2) s2n+2 < 10−10). Ïd·k
|Gi(z)| < 6 · (1 + 2 · sn+1/2)2 s2n+2 + 0.3 sn+1/2 < 0.4 sn+1/2. (3.7.17)
50 1nÙ Julia 8 Cantor ±kn¼ê
l , XJ z ∈ Ωi, K
|z2n+2 − b2n+2i | · |Gi(z)| < (2 + sn+1/2)|bi|2n+2 · 0.4 sn+1/2 < sn+1/2|bi|2n+2. (3.7.18)
â (3.7.12) Ú Rouche ½n, 3 Fn(z) = 0 ) wi,j ¦éz 1 ≤ j ≤ 2n+ 2
k wi,j ∈ Ωi. AO/,âÚn 3.2.1(2)1ã |wi,j− wi,j| < sn+1/2|bi|. d,
aqu (3.2.14) Ú (3.2.15) y, wi1,j1 = wi2,j2 = (i1, j1) = (i2, j2).
é 1 ≤ i ≤ n− 1, - CPi := wi,j : 1 ≤ j ≤ 2n+ 2 Pn “~C” ± T|bi|.:¤8Ü.
Ún 3.7.5. 3÷v An−1 ≺ · · · ≺ A1 n − 1 Ain−1i=1 ±9©O¹ 0
Ú ∞ üüëÏ« U0 Ú U∞, ¦
(1) U∞ ⊃ C \ D Pn(U∞) ⊂ U∞ ∪ 1;(2) Ai ⊃ T|bi| ∪ CPi, i Ûê Pn(Ai) ⊂ U0, i óê Pn(Ai) ⊂ U∞;
(3) n óêk Pn(U0) ⊂ U∞ n Ûêk Pn(U0) ⊂ U0.
y². - U∞ := C \D 4ü Ü. XJ·5¿Ún 3.7.2, K (1) ´w
,. - ε = sn+1/2 Ai = A|bi|(1−2ε),|bi|(1+2ε). â (3.7.1), ·
|Rn(z)| :=∣∣∣∣Pn(z)−Bn
An· nz
n+1 + 1
n+ 1
∣∣∣∣ = |z|(−1)n+1(n+1) |z2n+2− b2n+2i |(−1)i−1
Hi(z), (3.7.19)
Ù¥
Hi(z) =i−1∏j=1
|bj|(2n+2)(−1)j−1n−1∏j=i+1
|z|(2n+2)(−1)j−1 ·Qi(z) (3.7.20)
Qi(z) =i−1∏j=1
∣∣1− (z/bj)2n+2
∣∣(−1)j−1n−1∏j=i+1
∣∣1− (bj/z)2n+2∣∣(−1)j−1
. (3.7.21)
XJ z ∈ Ai, Ù¥ 1 ≤ i ≤ n− 1, ·k
Qi(z) <i−1∏j=1
(1 + 3 |bi/bj|2n+2
) n−1∏j=i+1
(1 + 3 |bj/bi|2n+2
)< (1 + 6s2n+2)2 (3.7.22)
Qi(z) >i−1∏j=1
(1 + 3 |bi/bj|2n+2
)−1n−1∏j=i+1
(1 + 3 |bj/bi|2n+2
)−1> (1 + 6s2n+2)−2. (3.7.23)
§3.7. õV CANTOR ±. JULIA 8~f 51
5¿ ε = sn+1/2 ≤ (5n)−2n−1 ≤ 10−5. XJ n ´óê 1 ≤ i ≤ n − 1 ´Ûê, K
é z ∈ Ai, ·k
|Rn(z)| = |z2n+2 − b2n+2
i ||z|n+1
1
s(i−1)(n+1)Qi(z) <
|bi|n+1(1 + (1 + 2ε)2n+2)
(1− 2ε)n+1
(1 + 6s2n+2)2
s(i−1)(n+1)
=1 + (1 + 2ε)2n+2
(1− 2ε)n+1(1 + 6s2n+2)2sn+1 < 2.1 · sn+1.
XJ n Ú 1 ≤ i ≤ n− 1 Ñ´óê, Ké z ∈ Ai, ·k
|Rn(z)| = |bi−1|2n+2|z|2n+2
|z|n+1|z2n+2 − b2n+2i |
1
s(i−2)(n+1)Qi(z) >
(1− 2ε)n+1
1 + (1 + 2ε)2n+2(1− 6s2n+2)2 > 0.49.
ùL²XJ n ´óê 1 ≤ i ≤ n− 1 ´Ûê, éu z ∈ Ai, âÚn 3.7.1(2), ·k
|Pn(z)| <∣∣∣∣2.1 · sn+1 · (n+ 1)An
nzn+1 + 1
∣∣∣∣+ |Bn|
≤ 2.1 (sn+1/2/5) · (1 + s2n+1/(n+ 1))
1− n(1 + 2ε)sn+1+
s2n+1
3n+ 3< sn+1/2.
XJ n Ú 1 ≤ i ≤ n− 1 Ñ´óê, Ké z ∈ Ai, ·k
|Pn(z)| >∣∣∣∣0.49(n+ 1)An
nzn+1 + 1
∣∣∣∣−|Bn| ≥0.49(n+ 1)(1− s2n+1/(n+ 1))
1 + n(1 + 2ε)sn+1− s2n+1
3n+ 3>n+ 1
3≥ 1.
|^aq?Ø, ±y²XJ n ´Ûê, éu z ∈ Ai, ·k
éÛê i k |Pn(z)| < sn+1/2 éóê i k |Pn(z)| > 1. (3.7.24)
- U0 = Dr, Ù¥ r = sn+1/2. ù (2) y².
XJ n ´Ûê, éz¦ |z| ≤ sn+1/2 z, ·k
|Pn(z)| ≤∣∣∣∣(n+ 1)Annzn+1 + 1
∣∣∣∣ |z|n+1
n−1∏i=1
|bi|(2n+2)(−1)i−1n−1∏i=1
∣∣∣∣1− z2n+2
b2n+2i
∣∣∣∣(−1)i−1
+ |Bn|
≤(n+ 1)(1 + s2n+1/(n+ 1))
1− nsn2+n/2s3(n+1)/2
n−1∏i=1
(1 + 2
|z|2n+2
|bi|2n+2
)+
s2n+1
3n+ 3< sn+1/2.
ùL²XJ n Ûê, K Pn(Dr) ⊂ Dr, Ù¥ r = sn+1/2.
XJ n óê, K Pn ò 0 N ∞ . éz¦ |z| ≤ sn+1/2
z, ·k
|Pn(z)| ≥ (n+ 1) s−(n+1)/2 (1− s2n+1/(n+ 1))
1 + nsn2+n/2
n−1∏i=1
(1− 2
|z|2n+2
|bi|2n+2
)− s2n+1
3n+ 3> n > 1.
(3.7.25)
ù (3) Òy². Úny..
52 1nÙ Julia 8 Cantor ±kn¼ê
ã 3.6 P3 Cantor ±. Julia 8. ëê s À¿©. ã¥ÚÜ©
L@ªS¹:@ Fatou ©|, xÚÜ©KL«@ªS
>.¹ØÄ: 1 Ô Fatou ©|. Ô Fatou ©|Ú§c¥
Fatou I³xÑ5. ã¡: [−1.6, 1.6]× [−1.2, 1.2].
½n 3.1.4 y². - A := C \ (U0 ∪ U∞). Pn Julia 8u⋂k≥0 P
−kn (A). 5¿
Pn ´AÛk. ?Øaqu½n 3.1.1 Ú 3.1.3 y². Pn Julia ©|¤
8ÜÓu n ÎÒ Σn := 0, 1, · · · , n− 1N ü>²£. AO/, Pn Julia 8Ó
u Σn × S1, ù´·@ Cantor ± (ã 3.6). [L§3dÑ.
§3.8 Ún 3.7.2 y²
ù!·y²Ún 3.7.2, TÚn´y²Ún 3.7.5 Ú½n 3.1.4 '.
y². - R(z) = 1/Pn(1/z),KÚn 3.7.28(y² R(D) ⊂ D∪1. - w = zn+1,
ÏLO, ·k
R(w) := R(z) =w + n
n+ 1· 1
S(w), (3.8.1)
Ù¥
S(w) = An
n−1∏i=1
(1− b2n+2i w2)(−1)i−1
+w + n
n+ 1Bn = 1 +
w − 1
1 + (2n+ 2)Cn
(H(w)− 1
w − 1+ 2Cn
)(3.8.2)
§3.8. Ún 3.7.2 y² 53
H(w) =n−1∏i=1
(1− b2n+2i )(−1)i
n−1∏i=1
(1− b2n+2i w2)(−1)i−1
. (3.8.3)
du H(1) = 1, ùL² H ′(1) ´kê. ¯¢þ,
I(w) :=H ′(w)
H(w)= −2w
n−1∑i=1
(−1)i−1b2n+2i
1− b2n+2i w2
. (3.8.4)
l I(1) = H ′(1) = −2Cn. é?Û¿© w − 1, ·±ò S(w) ¤
S(w) = 1 +(w − 1)2
1 + (2n+ 2)Cn·H(w)−1w−1
+ 2Cn
w − 1=: 1 +
(w − 1)2
1 + (2n+ 2)Cn· Φ(w), (3.8.5)
Ù¥
Φ(w) =∑k≥2
H(k)(1)
k!(w − 1)k−2. (3.8.6)
eÚ´é?Û k ≥ 2 O H(k)(1).
éz k ≥ 1, -
Yk(w) =n−1∑i=1
(−1)i−1
(b2n+2i
1− b2n+2i w2
)k. (3.8.7)
AO/, Y1(1) = Cn
Y ′k(w) = 2kw Yk+1(w). (3.8.8)
XJ |w| = 1, ·k
|Yk(w)| ≤∣∣∣∣ b2n+2
1
1− b2n+21
∣∣∣∣k(
1 +n−1∑i=2
∣∣∣∣b2n+2i (1− b2n+2
1 )
b2n+21 (1− b2n+2
i )
∣∣∣∣k)≤ 11
10
∣∣∣∣ b2n+21
1− b2n+21
∣∣∣∣k . (3.8.9)
aq/, ·k |Yk(w)| ≥ 910|b2n+2
1 /(1− b2n+21 )|k. ùL²∣∣∣∣Yk+1(w)
Yk(w)
∣∣∣∣ ≤ 11
9
∣∣∣∣ b2n+21
1− b2n+21
∣∣∣∣ ≤ 2s2n+2 < 1/2. (3.8.10)
·Äkäóéz k ≥ 0 k |I(k)(1)| ≤ 2k+1k!|Cn|. du I(0)(w) = −2wY1(w)
I(1)(w) = −2Y1(w)− 4w2Y2(w), ±8B/y² I(k)(w) ±¤
I(k)(w) =2k∑j=1
Qk,j(w) =2k∑j=1
Pk,j(w)Yk,j(w), (3.8.11)
Ù¥ Pk,j(w)´NÝõ k+ 1õªé, 1 ≤ l ≤ k+ 1k Yk,j = Yl. 5
¿, Qk,j Uu 0 (éAõª Pk,j NÝw¤´ −∞)Lª (3.8.11)
54 1nÙ Julia 8 Cantor ±kn¼ê
±z, ·IÒ´ù “” L«. AO/, Ø5, é 1 ≤ j ≤ 2k, ·
?Ú¦
Pk+1,2j−1(w)Yk+1,2j−1(w) = P ′k,j(w)Yk,j(w) Pk+1,2j(w)Yk+1,2j(w) = Pk,j(w)Y ′k,j(w).
(3.8.12)
5¿ deg(Pk,j) ≤ k + 1 é, 1 ≤ l ≤ k + 1 k Yk,j = Yl. ,, d (3.8.10), éz
l ≥ 1 k |Yl+1(1)/Yl(1)| ≤ 1/2. ùL²
|Pk+1,2j−1(1)Yk+1,2j−1(1)|+ |Pk+1,2j(1)Yk+1,2j(1)|
= |P ′k,j(1)Yl(1)|+ |Pk,j(1)Y ′l (1)|
≤ (k + 1)|Pk,j(1)Yl(1)|+ 2(k + 1)|Pk,j(1)Yl+1(1)|
≤ 2(k + 1)|Pk,j(1)Yk,j(1)|.
(3.8.13)
P ||I(k)(1)|| :=∑2k
j=1 |Pk,j(1)Yk,j(1)|, ·k ||I(k)(1)|| ≤ 2k||I(k−1)(1)||. ùL²
|I(k)(1)| ≤ ||I(k)(1)|| ≤ 2kk!||I(0)(1)|| = 2k+1k!|Cn|. (3.8.14)
ùÒy²éz k ≥ 0, äó |I(k)(1)| ≤ 2k+1k!|Cn| ¤á.
Ùg, ·8B/yé k ≥ 1 k |H(k)(1)| ≤ 4kk!|Cn|. é k = 1, ·k |H ′(1)| =2|Cn| < 4|Cn|. béz 1 ≤ i ≤ k k |H(i)(1)| ≤ 4ii!|Cn|. â (3.8.4), ·
k H ′(w) = H(w)I(w). Ï |I(k−i)(1)| ≤ 2k−i+1(k − i)!|Cn| éz 1 ≤ i ≤ k k
|H(i)(1)| ≤ 4ii!|Cn|, l
|H(k+1)(1)| ≤ |I(k)(1)|+k∑i=1
k!
i!(k − i)!|H(i)(1)| · |I(k−i)(1)|
≤ 2k+1k!|Cn|(1 + 2k+1|Cn|) ≤ 4k+1(k + 1)!|Cn|.
(3.8.15)
XJéu |θ| ≤ 1/20 k w = eiθ, K |w − 1| < |θ| ≤ 1/20. â (3.8.6) Ú (3.8.15),
·k
|Φ(w)| ≤∑k≥2
4k|Cn|(1/20)k−2 ≤ 16|Cn|∑k≥0
5−k = 20|Cn|. (3.8.16)
â (3.8.5) Ú (3.8.16), k
|S(w)| ≥ 1− θ2
1− (2n+ 2)|Cn|20|Cn| ≥ 1− s2n+1
n+ 1θ2. (3.8.17)
ù´du n ≥ 2 â (3.7.6) k |Cn| < s2n+1/(8(n+ 1)2).
§3.8. Ún 3.7.2 y² 55
,¡, XJé 0 ≤ |θ| ≤ π k w = eiθ, K∣∣∣∣w + n
n+ 1
∣∣∣∣ =
(1− 4n
(n+ 1)2sin2 θ
2
)1/2
≤(
1− 4n
π2(n+ 1)2θ2
)1/2
≤ 1− 2n
(n+ 1)2π2θ2.
(3.8.18)
ù´Ïé 0 ≤ x < 1 k (1− x)1/2 ≤ 1− x/2. ùL²XJé |θ| ≤ 1/20 - w = eiθ, K
|R(w)| ≤ (1− 2n
(n+ 1)2π2θ2)(1− s2n+1
n+ 1θ2)−1 ≤ 1. (3.8.19)
d, |R(w)| = 1 = w = 1.
XJé |θ| > 1/20 k w = eiθ, â (3.8.2) ÚÚn 3.7.1(2), ·k
|S(w)| ≥ (1− s2n+1
n+ 1)n−1∏i=1
(1− |bi|2n+2)− s2n+1
3n+ 3≥ 1− 3s2n+1
n+ 1. (3.8.20)
â (3.8.18) Ú (3.8.20), ·k
|R(w)| ≤ (1− 2
202(n+ 1)π2)(1− 3s2n+1
n+ 1)−1 < 1. (3.8.21)
ùL²Ø w = 1ù:, R(w)òü >.Nü . ÏXJ |w| ≤ 1
K R(w) 6= ∞, · R(D) ⊂ D ∪ 1. Ïd, R(D) ⊂ D ∪ 1 R òk8
z ∈ C : zn+1 = 1 N÷ 1. ùÚn 3.7.2 Òy².
1oÙ McMullen N Julia 8[é¡x
éukn¼ê Julia 8ÿÀ5, éõö®²ïÄ. éukn¼ê Julia
8[é¡AÛ5, Kék<9. ¯¢þ, =B´éu~AÏ/, 8cé
k<é Julia 8[é¡AÛ5?1x. ùÙ¥, ·òÄ McMullen N
Julia 8[é¡AÛ5. ·Ñgd.;<º¹e[é¡üz©
a, ¿égd.:;<º¹?1?Ø. AO/, ·y²3Vkn
¼ê, Ù Julia 8[é¡du/#.
§4.1 cóÚĽÂ
- l,m ≥ 2 ü÷v 1/l + 1/m < 1 ê, I0 = [0, 1] ⊂ R ¢¶þ4ü «m. ·ò I0 lm©¤ 3 f«m, z«mÝ 1/l, 1 − 1/l − 1/m, 1/m,
,ò¥m«mSÜK.#8Ü I1 ü«m¿ [0, 1/l]∪ [1−1/m, 1].
- g0(x) = (1− x)/l g1(x) = 1 + (x− 1)/m. ·k I1 = g0([0, 1]) ∪ g1([0, 1]). éz
n ≥ 1,- In =⋃i1,··· ,in∈0,1 gi1 · · · gin([0, 1]). Kz In Ñ´;k In ⊇ In+1. ·
¡¤k In ⋂n≥0 In ´IO Cantor 8,¿P Cl,m. AO/, C3,3 IOn
© Cantor 8. ½ÂIO Cantor ± Al,m Cl,m×S1, Ù¥ S1 := z ∈ C : |z| = 1´ü ±.
S ⊂ C ´ Riemann ¥¡þëÏÛÜëÏ;8. XJ S SÜ´8,
C \ S z©|>.Ñ´ Jordan C \ S ØÓ©|4*dpØ, K
¡ S ´ Sierpinski /# (k¡/#) [74]. d/, XJ S SÜ´8
±¤ S = C \⋃i∈NDi, Ù¥ Di ´÷vXJ i 6= j K ∂Di ∩ ∂Dj = ∅ Jordan
«, d, i → ∞ ¥¡» diam(Di) → 0, K S ´ Sierpinski /#. ¤k
Sierpinski /#*dÑ´Ó. XJéz i ∈ N, Di ¥¡ (½öî¼) ±, K
ù/#¡´/#.
XJ®²üÝþm (X, dX) Ú (Y, dY ) ´Ó, įK´wùü
m´Ä´*d[é¡d. - (X, dX) Ú (Y, dY ) üÝþm. XJ3 X
Y Ó f : X → Y Ú η : [0,∞)→ [0,∞) ¦éuzØÓ: x, y, z ∈ X k
dY (f(x), f(y))
dY (f(x), f(z))≤ η
(dX(x, y)
dX(x, z)
),
K (X, dX) Ú (Y, dY ) ¡´*d[é¡d. XJ X = Y = C, K X Y Ó
´[é¡=§´[/ ( [36, ½n 11.14]).
56
§4.1. cóÚĽ 57
ùÙ¥, ·ïÄ McMullen N
fλ(z) = zm + λ/zl, (4.1.1)
Julia 8[é¡AÛ, Ù¥ λ ∈ C∗ := C \ 0 l,m ≥ 2 ü÷v 1/l+ 1/m < 1
ê. N´wÑ fλ 3 ∞ ká5ØÄ:, Ù_ 0 Ú§gC. k, XJ·
rNê l Ú m Ø´ λ, K^ f l,m Ú J l,m 5©OL« fλ Ú§éA Julia 8
Jλ. Riemann ¥¡ C ;f8¡´ Cantor ±XJ3ù;8ÚIOn
© Cantor 8 A3,3 mkÓ.
éu Jλ ÿÀ5, Devaney, Look Ú Uminsky y²<ºn©½n. ù½
n`XJ fλ ¤k.:Ñ ∞ áÚ, K Jλ o´ Cantor 8, Cantor ±
½ö´ Sierpinski /# ( [24, ½n 0.1] ½ö½n 4.2.1). éu Jλ AÛ5, ·
ke¡½n.
½n 4.1.1. b fλ ¤k.:Ñ ∞ áÚ. Ke¡n«¹k=k«u
):
(1) Jλ ´ Cantor 8, K Jλ [é¡duIOn© Cantor 8 C3,3;
(2) Jλ ´ Cantor ±, K Jλ [é¡duIO Cantor ± Al,m, Ù¥ l
Ú m (4.1.1) ¥ê;
(3) Jλ ´ëÏ, K Jλ [é¡du/#.
Ø 0 Ú ∞, N fλ e l + m .:¡´gd.:. ùgd.
:þkÓ; ( §4.2). §oÓáÚ ∞ o;Ók.. XJgd
.:vk<º ∞, K Jλ ´ëÏ ( [25]). 3ù«¹e, üg,¯KÑy:
(1) Jλ oÿ´ Sierpinski /#? (2) §´Ä[é¡du/#?
XJgd.;k.á5±Ï;¤áÚ, § Fatou ©|÷v,«
“ìv” 5, K Jλ ´ Sierpinski /# ( [23]). éõ λ ¦ Jλ ´
Sierpinski /#, ·Ä fλ ëêm. ½Âùx¼ê<º,
Λ = λ ∈ C∗ : fλ gd.;´k.. (4.1.2)
®² Λ ´ëϹ Mandelbrot 8éõÓ, ù Mandelbrot 8Ó
¥:éAuëê ( [20, 69]).
P Mandelbrot 8 M := c ∈ C : Pc(z) = z2 + c Julia 8´ëÏ. M V©|´ M ëÏmf8, 3p¡ 0 Pc á5±Ï;¤áÚ.
- H M V©|. éz c ∈ H, õª Pc k±Ï k á5±Ï;.
d H ¡´ M ±Ï k V©|, ½öò H ±ÏP p(H). 3
58 1oÙ McMullen N Julia 8[é¡x
ëê rH á3 H >.þ¦õª PrH k¹.: 0 ±Ï k Ô
Fatou©|. ù: rH ¡´ H .AO/,- H♥ M ±Ï 1ëÏ©|. e
I ♥ x H♥ /G, §>.´^%9. M V©|±©¤üa: satellite
Ú primitive ( §4.3 ¥½Â). */, ùüa©|Ä«O´: primitive V
©|>.d%9¤, satellite V©|KØ´.
Ï Λ¥zV©| H Ñá3 Mandelbrot8,Óp ( [69,½
n 10] ڽ 4.3.2), é H, ·±½ÂéA rH ¿±ò Λ ¥V©|©
satellite Ú primitive ùü«a. ( §4.3).
éu Jλ oÿ´ Sierpinski /#¯K, [62] ¥é λ > 0 /Ä.
P Kλ = C \ A(∞) fλ “W¿” Julia 8, Ù¥ A(∞) ∞ áÚ.
½n 4.1.2 ([62, 76]). 3ü¢ê λ0 < λ1 ¦XJ λ ∈ [λ0, λ1], K fλ Julia
8 Jλ ´ÛÜëÏ. d, Jλ ´ Sierpinski /#=e¡÷v:
(I) Kλ SÜ Int(Kλ) = ∅ λ 6= λ0;
(II) λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿
H Ø´ H♥ 3ÓNe.
l Mandelbrot 8 Λ ÓN°(½Âò3 §4.3 ¥Ñ. [62] ¥=éþ
¡ã½n3 l = m ≥ 3 ¹y². Xie òù(J3¨Æ¬Ø©¥í
2/, y²g´aq ( [76]). ùÙ¥, ·òy²
½n 4.1.3. XJ λ ∈ [λ0, λ1], K Jλ [é¡du/#= λ ÷v (I)
½ö (II) λ 6= rH. AO/, XJ λ ∈ (λ0, λ1) ¦ fλ Ù¥.:´îý±Ï
, K Jλ [é¡du/#.
á=íØ, ·k
íØ 4.1.4. 3Vkn¼ê, Ù Julia 8[é¡du/#.
¯¢þ, â½n 4.1.3 y², ·$3ákn¼ê, Ù Julia
8[é¡du Sierpinski /#.
·òÄ/¦ Jλ ´ Sierpinski /#. AO/, ëê λ ±3E
²¡þCz Ø==´3¢¶þ. , , ù¦·b l = m ≥ 3.
½n 4.1.5. b l = m ≥ 3 fλ gd.;´k.. XJ λ ÷ve¡^
, K Jλ ´ Sierpinski /#.
(a) SÜ Int(Kλ) = ∅ gd.:;4 ∞ áÚ>.Ø;
§4.2. gd.:<º/ 59
(b) λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿
H Ø´ H♥ 3ÓNe.
d, XJ λ ÷v (a) ½ö (b) λ 6= rH, K Jλ [é¡du/#.
ùÙ´ùSü: 3 §4.2 ¥, ·ã McMullen NÄ5¿y²
½n 4.1.1. 3 §4.3 ¥, ·y²½n 4.1.3, íØ 4.1.4 Ú½n 4.1.5. 3 §4.4 ¥, ·
Ä Cantor ±[é¡AÛ.
§4.2 gd.:<º/
N fλ 3:kê l − 1 .:, 3 ∞ ákê m− 1
.:, ωj = e2πij/(l+m)(λl/m)1/(l+m) ´ü.:, Ù¥ 0 ≤ j ≤ l + m − 1. ù.
: ωjl+m−1j=0 ¡´ fλ gd.:. ÏL/O, |f nλ (ωj)| = |f nλ (ωk)| é
u 0 ≤ j, k ≤ l +m− 1 Ú n ≥ 0 ¤á. Ïdù l +m gd.:;þdÙ¥
:;û½.
§4.2.1 <ºn©½nÚëêm©)
- Bλ Ú Tλ fλ ©O¹∞Ú 0 Fatou©|. K Bλ∩Tλ = ∅½ö Bλ = Tλ.
Devaney, Look Ú Uminsky 3e¡½n¥Ñ McMullen N Julia 8ÿÀa
.©a½n.
½n 4.2.1 ([24]). b fλ gd.:Ñ<º ∞. Ke¡n«¹k=k
«u):
(1) 3, j ¦ fλ(ωj) ∈ Bλ, K Jλ ´ Cantor 8;
(2) 3, j ¦ fλ(ωj) ∈ Tλ 6= Bλ, K Jλ ´ Cantor ±;
(3) 3, j Ú k ≥ 1 ¦ f kλ (ωj) ∈ Tλ 6= Bλ, K Jλ ´ Sierpinski /#.
¯¢þ, fλ ëêm C∗ ±©¤oÜ© Λ t Λ∞ t Λ0 t ΛS ( [69, §7]), Ù
¥ Λ (4.1.2) ¥½Â<º,; Λ∞ = λ ∈ C∗ : Jλ ´ Cantor 8 ´Ã.«, ¡ Cantor ,; Λ0 := λ ∈ C∗ : Jλ ´ Cantor ± ´ 0 B
, ¡´ McMullen , Ï McMullen kïÄù«¹ ( [49, §7]);
ΛS = λ ∈ C∗ : Jλ ´ Sierpinski /# \ Λ ¡´ Sierpinski É, §dêõü
ëÏ©||¤. d, Λ ¥zV©|ѹ3 Mandelbrot 8,Ó¥ (
ã 4.2 ¥ã).
60 1oÙ McMullen N Julia 8[é¡x
§4.2.2 Cantor 8[é¡üz
David Ú Semmes ·K [17, ·K 15.11] y²: XJVkn¼ê
Julia 8ÓuIOn© Cantor 8 C3,3, Kù Julia 8[é¡du C3,3. â½n
4.2.1(1), ½n 4.1.1(1) ¤á.
§4.2.3 Cantor ±[é¡üz
·ò^[/Ãâ5y²½n 4.1.1(2). éü÷v 1/l + 1/m < 1 ê
l,m ≥ 2, ·3þ½ÂS¼êXÚ (P IFS), ¦ùS¼êXÚ
áÚf/duIO Cantor ± Al,m. ,·òùS¼êXÚ_ò
ÿl C g[KN. , 3é C þ3T[KNeØC Beltrami Xê¿nAE(, ·y²3[/NòùIO
Cantor ± Al,m N McMullen N f l,mλ Julia 8, Ù¥ëê λ 6= 0 ~.
½ 0 < r0 < 1,½Â I = [r0, 1]. - I0 = [r0, r0+(1−r0)/l] I1 = [1−(1−r0)/m, 1]
I üf«m. ½Â g0 : I → I0 g0(r) = (1− r)/l + r0 g1 : I → I1 g1(r) =
1− (1− r)/m. N´wÑS¼êXÚ g0, g1áÚfIO Cantor8 Cl,m 3
Ce. - A := reit : r ∈ I 0 ≤ t < 2π, Aj := reit : r ∈ Ij 0 ≤ t < 2πn, Ù¥ j = 0, 1. ½Â F : A0 t A1 → A ¦
F |A0(reit) = g−1
0 (r) · e−ilt F |A1(reit) = g−1
1 (r) · eimt.
5¿ F |A0 : A0 → A Ú F |A1 : A1 → A ©ONÝu l Ú m CXN. S¼
êXÚ (F |A0)−1, (F |A1)
−1 áÚf/duIO Cantor ± Al,m.
½Â F (z) = rl0/zl, §ò Dr0 := z : |z| < r0 N÷4ü Ü C \ D
F (z) = zm ò C \ D N÷gC. e5, ·òÿ F ¦ F ò Ar1,r2 := z :
r1 < |z| < r2 N÷ Dr0 , Ù¥ r1 = r0 + (1− r0)/l r2 = 1− (1− r0)/m.
XJ©|CX f : C → C ±¤ ψ R ϕ, Ù¥ R ´knN ψ, ϕ
þ[/Ó, K¡ f ´[K. - Tr = z : |z| = r ±:%» r > 0
±.
Ún 4.2.2. - f1(z) = r0rl1/z
l Ú f2(z) = r0zm/rm2 ü©O½Â3 Tr1 Ú Tr2
þCXN, ¦ f1(Tr1) = Tr0 f2(Tr2) = Tr0. K3ëYòÿ F : Ar1,r2 →Dr0 ¦ (1) F |Tri = fi, Ù¥ i = 1, 2; (2) F : Ar1,r2 → Dr0 ´[K; (3)
F (e2πi/(l+m)z) = e2mπi/(l+m)F (z).
y². Ù/ãEL§, ·=é m = 3 Ú l = 2 ¹y². Ù§¹
±aq/?Ø. ·Äkb½ r0 = r2 = 1 Ú r1 = ε ~, ÏÚn¥éòÿ
±3·EÐùbeòÿ2»?n=.
§4.2. gd.:<º/ 61
- A D ¥±:%Ê(«, ¿¦ A ⊃ Dε, Ù¥ xi, 0 ≤ i ≤ 9
ùÊ(U_ü 10º: x0 Ë 0. éu 1 ≤ j ≤ 5,- Bj ¹
> [x2j−2, x2j−1] Ú [x2j−1, x2j] Ê>/, Ù¥ x10 = x0. Ï ε v, ·±ÀJ
· A ¦ Dε ⊂ A∪⋃5j=1 Bj ⊂ D. éu 1 ≤ j ≤ 5, - y2j−2, y2j−1 Bj ,
üvkIPU_üº:. éu 1 ≤ j ≤ 5 Ú 2j − 2 ≤ k ≤ 2j − 1, - γk+j =
[yk, e2π(k+j)i/15] ë yk Ú e2π(k+j)i/15 ã. éu 0 ≤ n < 5, - γ3n = [x2n, e
2πni/5].
ùL²·k 15 ^ã γs ò E := D \ (A ∪⋃5j=1 Bj) U_^Sy©¤ 15 ÿÀ
o>/ Es ¿¦ γs Ú γs+1 Es ü^>, Ù¥ 0 ≤ s < 15 γ15 := γ0. aq/, -
ηt = [xt, εe2πti/10] ë xt Ú e2πti/10 ã, Ù¥ 0 ≤ t < 10. Kù 10 ^ã ηt ò
D := A \ Dε U_^Sy©¤ 10 ÿÀo>/ Dt ¿¦ ηt Ú ηt+1 Dt ü^
>, Ù¥ 0 ≤ t < 10 η10 := η0.
- B D ¥±:%Ê>/, Ù¥ zn, 0 ≤ n < 5 B ±_^S
ü 5 º:, Ù¥ z0 Ë 0. - ζn = [zn, e2πni/5] ë zn Ú e2πni/5 ã,
Ù¥ 0 ≤ n < 5. u´ù 5 ^ã ζn ò D \ B U_^Sy© 5 ÿÀo>/ Gn
¿¦ ζn Ú ζn+1 Gn ü^>, Ù¥ 0 ≤ n < 5 ζ5 = ζ0 (ã 4.1).
x0 γ0
E0
γ1y0
E1
B1
y1
γ2E2
x1
D0
D1
x2
B2
z0
z1
z2
z3 z4
ζ0
ζ1
ζ2
ζ3
ζ4
G0
G1
G2
G3
G4
B
ã 4.1 lNÝu 5©|CXN«¿ã,Ù¥ m =
3 l = 2. ã¥ÑaqÎÒ.
·Äkòÿ F : E0 → G0. E0 >.d γ0, [x0, y0], γ1 Úl [1, e2πi/15] |¤,
G0 >.d ζ0, [z0, z1], ζ1 Úl [1, e2πi/5] |¤. ®² F 3l [1, e2πi/15] þ
½Â F (z) = z3. ·½Â F (γ0, [x0, y0], γ1) = (ζ0, [z0, z1], ζ1) n±éAº
:N. K F ±[/òÿ E0 SÜ¿¦ F (E0) = G0. aq/, ½
Â[/òÿ F (E1) = G1 F (E2) = G2 ¦ F (γ1, [y0, y1], γ2) = (ζ1, [z1, z2], ζ2)
F (γ2, [y1, x2], γ3) = (ζ2, [z2, z3], ζ3). â½Âé¡5, éu z ∈ Es, Ù¥ 3 ≤ s < 15,
62 1oÙ McMullen N Julia 8[é¡x
½Â F (z) = e6πi/5F (e−2πi/5z).
·±3 A \ Dε þ|^aqòÿ¦ F (D0) = G4 Ú F (D1) = G3. é
u 2 ≤ t < 10 Ú z ∈ Dt, ½Â F (z) = e−4πi/5F (e−2πi/5z). éu 1 ≤ j ≤ 5, Ï Bj Ú
B ÑÊ>/ F 3 Bj >.þ´N, Ï 3 Bi B g,
/òÿ. y3, ·©|CX F : D \ Dε → D, Ù¥ x0, x2, x4, x6 Ú x8
©Ü:. âE, F A??´ÛÜ[/, l F ´[KN. d,
F (e2πi/5z) = e6πi/5F (z).
y3·Kb r0 = r2 = 1 Ú r1 = ε ~. éz z = reiθ ∈ Ar1,r2 , Ù¥
r ∈ [r1, r2] θ ∈ [0, 2π), ½Â
F (z) =1
r0
F ([ε+1− εr2 − r1
(r − r1)]eiθ).
K F ´Ún¥Iòÿ. y²..
íØ 4.2.3. òÿN F : C→ C ´NÝ l + m [KNé¤
k z ∈ C ÷v F (e2πi/(l+m)z) = e2mπi/(l+m)F (z).
y². â F ½Â±y.
·K 4.2.4. 3[/N ϕ : C → C ¦ ϕ F ϕ−1 = fλ, Ù¥ fλ ´
Julia 8 Cantor ± McMullen N. AO/, ½n 4.1.1(2) ¤á.
y². 3 C \ D þ½ÂIOE( σ(z) = σ0. âE, C ¥A¤k:3f ^eÑS C \ D. XJé¤k n ≥ 0 k F n(z) 6∈ C \ D, ½Â σ(z) = σ0. Ä
K, 3 n ¦ F n(z) ∈ C \ D. u´·½Â σ(z) = (F n)∗(σ0)(z), ùL
«IOE(3 C \D þ.£. Ϥk;²L Ar1,r2 õg, ùL² σ
´ C þ F−ØCE(.
- ϕ : (C, σ) → (C, σ0) È[/Ó¦ ϕ(0) = 0 z → ∞ ϕ(z)/z → 1. â F E, F ò ∞ ±NÝ m N ∞ 0 4:, ê
l. Ïd ϕ F ϕ−1 ´NÝu l +m kn¼ê. ùL² ϕ F ϕ−1 äk/ª
ϕ F ϕ−1 = (z − a1)(z − a2) · · · (z − al+m)/zl,
Ù¥ ajl+mj=1 F ":3 ϕ e.
âíØ 4.2.3 ¥ãé¡5, ajl+mj=1 þ!/á3±:%±
þ. Ïd ϕ F ϕ−1 = fλ, Ù¥ fλ(z) = zm + λ/zl ´ McMullen N λ =
(−1)l+ma1 · · · al+m é. y²..
§4.2. gd.:<º/ 63
§4.2.4 Sierpinski /#[é¡üz
é?Ûf8 X ⊂ C, X ¥¡»½Â diam(X) = supx,y∈X |x− y|, Ù¥ | · | L«¥¡Ýþ. é?Ûf8 X, Y ⊂ C, - dist(X, Y ) := infx∈X,y∈Y |x − y| X, Y m
¥¡ål.
½Â 4.2.5. γ ⊂ C ´^ Jordan , XJ3~ê k ≥ 1 ¦
diam(I) ≤ k |x− y| (4.2.1)
é¤kØÓ: x, y ∈ γ ¤á, Ù¥ I γ \ x, y äkᥡ»©|. @où
γ ¡´[. d, ¡ γ ´ k k.ò=.
γ ´^ Jordan , XJ§´ü ±3l C g[/N,
K γ ´[. XJù[/N ûØL K, @où^ γ ¡´
K–[. d, ~ê K Ú k *dp6.
½Â 4.2.6. XJéz i ∈ N, γi ´ K–[, Ù¥ K ´Ø6u i ~
ê, Kùx Jordan γii∈N ¡´x[. XJ3~ê s > 0 ¦
dist(γi, γj)
mindiam(γi), diam(γj)≥ s, (4.2.2)
Ù¥ i, j ∈ N i 6= j, Kùx Jordan γii∈N ¡´©.
½n 4.2.7 ([9, ½n 1.1]). b Γ := γii∈N ´ C ¥x Jordan , §©
OxpØ4 Jordan «>.. XJ Γ ´dx[¤¿Ù¥
*d´©, K3[/N f : C → C ¦é?¿ i ∈ N, f(γi)
´¥¡±.
y²½n 4.1.1(3),·Iy²XJgd.:´<º Jλ ´ Sierpinski
/#,K fλ Fatou©|>.¤8Ü´x[§÷vé©^
.
Ún 4.2.8 ([50, ½n 2.5]). - A ⊂ C äkØ γ, Ù mod(A) >
m > 0. - D C \ A k.©|. K3î¼Ýþe,
dist(D, γ) > C(m) diam(γ),
Ù¥ C(m) > 0 ´=6u m ~ê.
64 1oÙ McMullen N Julia 8[é¡x
þãÚnL²XJ Ae.±,@o dist(D,C\A ∪D)/diam(D)
e.±.
½n 4.2.9. b fλ gd.:´<º Jλ ´ Sierpinski/#. - γii∈NL« fλ ¤k Fatou ©|>.¤8Ü. K3~ê K ≥ 1 ¦z γi ´
K–[ γii∈N ÷v (4.2.2).
y². XJ fλ ¤kgd.:Ñ<º Jλ ´ëÏ, ÏL[/Ãâ,
± ∞ áÚ>. ∂Bλ ´[. Ïd ∂Bλ eZg_, z
γi Ñ´[ (5¿d¤k Fatou ©|Ñ´üëÏ).
3 C \ Bλ ¥ÀJ¿©C ∂Bλ Jordan ζ, ¦± ζ Ú ∂Bλ >.
« A0 ع fλ .;. ù´±Ï¤k.:Ñ<º ∞. u
´ A0 c A1 := f−1λ (A0) ´, Ùk.Ö©| Tλ.
/, é?¿ i ≥ 0, ·^ γi,l1≤l≤ki 5L« f−iλ (∂Bλ) ¤këÏ©|, Ù¥
ki 6u i ê. AO/, k0 = k1 = 1, γ0,1 = ∂Bλ γ1,1 = ∂Tλ. du A0 Ø
¹ fλ .;, é?¿ i ≥ 1, f−iλ (A0) ?ÛëÏ©|Ñ´. ·^ Ai,l
5L« f−iλ (A0) ¹ γi,l ëÏ©|.
â½n 4.2.1(3),3 k ≥ 1¦¤kgd.:ÑS Tλ. Ï e i ≥ k+1,
K fλ 3 Ai,l Ú§k.Ö©|þ´/. u´3~ê C1 ≤ C2 ±9 ρ < 1,
¦é? i ≥ 0 Ú 1 ≤ l ≤ ki, k
C1ρi ≤ diam(γi,l) ≤ C2ρ
i.
- N ≥ 1 ¦ C2ρN < C1 ¤áê. À ζ ¿©C ∂Bλ, yXJ
(i, l1) 6= (j, l2), K Ai,l1 ∩ Aj,l2 = ∅, Ù¥ 0 ≤ i, j ≤ N , 1 ≤ l1 ≤ ni, 1 ≤ l2 ≤ nj. 5¿
ù(Øé p ≤ i, j ≤ p+N ¤á, Ù¥ p ≥ 0.
du i ≥ k + 1 , fλ 3 Ai,l Ú§k.Ö©|þ´/. Ïd, 3~
ê m > 0 ¦é¤k i ≥ 0 Ú 1 ≤ l ≤ ni k mod(Ai,l) ≥ m. âíØ 2.4.2, ùL²
γii∈N [.
Ai,l1 Ú Aj,l2 A0 ÏL¦eZg_ØÓ.Ø5,b i ≥ j.
e i− j ≤ N , K Ai,l1 ∩ Aj,l2 = ∅. - βi,l1 Ú βj,l2 ©O Ai,l1 Ú Aj,l2 Ø. âÚ
n 4.2.8 , 3~ê C(m) ¦
dist(γi,l1 , γj,l2)
mindiam(γi,l1), diam(γj,l2)≥ min
dist(γi,l1 , βi,l1)
diam(βi,l1),dist(γj,l2 , βj,l2)
diam(βj,l2)
≥ C(m).
§4.3. gd.:<º/ 65
e i− j > N , Kd N À diam(γi,l1) < diam(γj,l2). 5¿d Ai,l1 Aj,l2
U, , ·%k Ai,l1 ∩ γj,l2 = ∅. âÚn 4.2.8
dist(γi,l1 , γj,l2)
mindiam(γi,l1), diam(γj,l2)=
dist(γi,l1 , γj,l2)
diam(γi,l1)≥ dist(γi,l1 , βi,l1)
diam(βi,l1)≥ C(m).
ùL² γii∈N ÷vé©^.
½n 4.1.1 y². (1) ´â David Ú Semmes ·K ( [17, ·K 15.11]). (2)
¤á´Ï·K 4.2.4. (3) (ܽn 4.2.7 Ú 4.2.9.
§4.3 gd.:<º/
P M Mandelbrot 8. gõª Pc(z) = z2 + c, c ∈M ¡´V=§kk.áÚ±Ï;. é?Û½ M V©| H Ú§ rH , PrH k
±Ï m Ô±Ï;. ®²3,ê v ≥ 1 ¦ p(H) = vm (
[52, Ún 6.3]), Ù¥ p(H) V©| H ±Ï. XJ v = 1, K H ¡´ primitive
, XJ v ≥ 2, K§¡´ satellite . */, M V©|´ primitive
=Ø3Ù§V©|N3 H þ. ½ö`, H primitive =§
>.´%9.
§4.3.1 nØ
·Äk£Ák'nØ, §±3 [29, 50] ¥é. b U, V Cþ÷v U ⊂ V üëÏ«. XJ g : U → V ´X_;N, K¡n| (g, U, V )
´aõª. (g, U, V ) W¿ Julia 8½Â K(g) :=⋂n≥0 g
−n(U). 5¿3W¿
Julia 8þ, g ?ÛgSÑk¿Â. gõª Pc ¡´XJ3¹ 0
U, V ¦ (P nc , U, V ) ´aõª, Ù¥ n > 1 ±Ï. - Kj := P jc (K0),
K Pc(Kj) = Kj+1, Ù¥ j = 0, 1, · · · , n− 1 Kn = K0. z Kj Ñ¡´
Julia 8§þ Pc Julia 8 JPc f8. XJ i 6= j, K Ki Ú Kj SÜ´Ø
, Ù¥ 0 ≤ i, j < n. Ý 0 3 JPc þXº:´½5½öÔØÄ
:, ¡ Pc β– ØÄ:. ®² β–ØÄ: Julia 8´Ø [50, ½n 7.10].
XJé 0 ≤ i, j < n k Ki ∩Kj = ∅, Kù«¡´Ø..
e¡(J´¬m½n, ±l [52, ½n 2.4, Ún 2.7Ú §6]íÑ.5¿ H♥
M ±Ï 1 V©|.
½n 4.3.1. - H M V©| H 6= H♥. é? c ∈ H, K
(1) Pc ´, ±Ï p(H).
66 1oÙ McMullen N Julia 8[é¡x
(2) ´Ø.= H ´ primitive . AO/, Pc k. Fatou ©|
4*dpØ= H ´ primitive , Ù¥ c ∈ H ∪ rH rH H
.
§4.3.2 ÓN
£½Â3 (4.1.2) ¥ Λ fλ <º,. d [69, §7] Λ zV©|
M V©|3 Λ ¥Ó. - ΦM : M →M Mandelbrot 8Ú§3 Λ p¡
mÓ, ¦ ΦM ò M zV©| H N÷ Λ ,V©|. ù
Ó¦÷ve¡ü^: (1) XJ c ∈ H ¦ Pc k¦f ζ ∈ D áÚ±Ï;, K Φ(c) ∈ H ´¦ McMullen N fΦ(c) k¦f ζ á5±Ï;
ëê, Ù¥ H = ΦM(H) M V©|. H ½Â rH = ΦM(rH). (2) Ó
ΦM : M →M ´. äN/, XJ ΦM′ : M →M′ ´÷v^ (1) Ó,
KM′ ⊂M.
½Â 4.3.2. ÓM ¡´ Mandelbrot 83 Λ ¥E (copy)
N ΦM ¡´ÓN.
é Λ¥zV©|H,3 Λ¥MandelbrotE¹H. V
©| H ¡´ satellite½ primitiveXJ H 3ÓNe_´ satellite
½ö primitive .
éu λ ∈ C∗, ½Â fλ “W¿” Julia 8
Kλ := z ∈ C : f kλ (z)k≥0k..
XJ λ ∈ C∗ \ Λ, K Kλ = Jλ. XJ λ ∈ Λ, KU3¹3 Kλ ¥ Fatou ©|
ªØ¬SááÚ Bλ ¥. ½n 4.3.1 íØ, e¡Únd
[29] ¥a Mandelbrot 8nØíÑ.
Ún 4.3.3. - H Λ ¥E M = ΦM(M) ¦ H 6= ΦM(H♥) V
©|. é?Û λ ∈ H∪ rH, Ù¥ rH H , W¿ Julia 8SÜ Int(Kλ) ©|
4*dØ= H ´ primitive .
§4.3.3 <º¹e Sierpinski /#[é¡üz
P λ0 = supλ : λ ∈ Λ0 ∩ R+ λ1 = infλ : λ ∈ Λ∞ ∩ R+, Ù¥ Λ0 Ú Λ∞ ©O
L« McMullen Ú Cantor ,.
½n 4.3.4. XJ λ ∈ [λ0, λ1], K Jλ [é¡du/#=e¡ö
¤á:
§4.3. gd.:<º/ 67
(I) W¿ Julia 8SÜ Int(Kλ) = ∅ Ú λ 6= λ0;
(II) λ ∈ H, Ù¥ H ´ M 3 Λ ¥Ó primitive V©|¿ HØ´ H♥ 3ÓNe.
y². â½n 4.1.2, ·Iy²¿©Ü©`²XJ λ = rH, K Jλ ØU[é
¡du/#, Ù¥ H ´ Λ primitiveV©|. XJ H ´ primitive
V©|, @o JrH ØU[é¡du/#´w,Ï frH Fatou 8¹
kk:üëÏÔ Fatou ©|, ù©|>.½Ø´[.
â [62,Ún 4.1]½ö [76,Ún 4.15],3lMandelbrot8N Λ,f8
ÓN Φ : M →M,¦M∩R+ = [λ0, λ1]. AO/, Φ([−2, 1/4]) = [λ0, λ1].
d, éz c ∈ [−2, 1/4], 3½Â3 Pc Julia 8 JPc S[
/Ó Ψc ¦ Ψc(JPc) ⊂ KΦ(c) Ψc(JPc) ∩ R = [pΦ(c), qΦ(c)], Ù¥ 0 < pλ < qλ ü
¢ê¦ λ ∈ [λ0, λ1] fλ(pλ) = fλ(qλ) = qλ. ¯¢þ, pλ = supz : z ∈ Tλ ∩ R+ qλ = infz : z ∈ Bλ ∩ R+ (ã 4.2).
- P (fλ) := Closuref kλ (ωj) : 0 ≤ j < l +m, k > 0 fλ (gd) postcritical 8
4. dþã©Û, ·éz λ ∈ (λ0, λ1], Tλ _zëÏ©|4
¢¶Ø. ùL²éu f−1λ (T λ) zëÏ©| U ′, Ñ3 U ′ üëÏm
U , ¦ P (fλ) ∩ U = ∅.XJ Int(Kλ) = ∅, KØ Bλ , ¤k Fatou ©|ÑS Tλ þ. Ï ∂Bλ
´[ ( [62, Ún 3.8] ½ [76, Ún 4.14]), ÏLaqu½n 4.2.9 ?
Ø, ±`² Jλ ÷v[Ú©^. ùL²XJ Int(Kλ) = ∅ λ 6= λ0, K
Jλ [é¡du/#.
XJ λ ∈ H ¦ Jλ ´ Sierpinski /#, K fλ ´V. aqu½n 4.2.9
?رy² Jλ [é¡du/#. y²..
½n 4.1.3 ÚíØ 4.1.4 y². §´½n 4.3.4íØ (ã 4.2m>ã
).
e5, ·o´b½ n := l = m ¦ fλ NÝ´ 2n, Ù¥ n ≥ 3 ´½
ê. e¡½n´ [61] ¥½n 1.3 f.
½n 4.3.5 ([61]). b Julia 8 Jλ ´ëÏ fλ .;ر>. ∂Bλ þ
:à:, K Jλ ´ÛÜëÏ.
- P Ú Q ©OL«¹3 C \Kλ Ú Kλ ¥ Fatou ©|¤8Ü.
68 1oÙ McMullen N Julia 8[é¡x
ã 4.2 N fλ(z) = z3 + λ/z3 ëêmÚÄåXÚ²¡. ÀJëê λ =
0.0258324407020185 · · · ¦ fλ ¤kgd.:Ñ´ý±Ï Julia 8 Jλ
´ Sierpinski /#¿[é¡du/#.
Ún 4.3.6. - U ∈ Q ¹3 Kλ ¥ Fatou ©|. XJ λ ∈ H ∪ rH, Ù
¥ H ´ Λ ¥ primitive V©|, H Ø´ H♥ 3ÓNe, K
U ∩Bλ = ∅.
y². b U ∩Bλ 6= ∅. XJI, Ä fλ eZgEÜ, ·±b½ U 3
fλ e´ØC. Äk, ·äó U ∩ Bλ = z0 ´ fλ ØÄ:. du ∂Bλ ´
^ Jordan [61, ½n 1.1], 3Ó γ : S1 → ∂Bλ ¦ γ(eint) = fλ(γ(eit)). XJ
](U ∩ Bλ) ≥ 2, K3 t1 < t2 ¦ γ(eit1), γ(eit2) ∈ U ∩ Bλ U ∩ Bλ ⊂ γ([eit1 , eit2 ]),
Ù¥ 0 ≤ t2 − t1 ≤ 2π/n, ù´Ï fλ Fatou ©|äké¡5. Ï U ´ØC, K
fλ(γ([eit1 , eit2 ])) = γ([eit1 , eit2 ]). , ,é?Ûfl I ⊂ Bλ,3 k ≥ 0¦ f kλ (I) = ∂Bλ,
ù´Ï ∂Bλ ´^ Jordan fλ 3 ∂Bλ þÝu z 7→ zn. ù´gñ, l
äóy².
â [69, §7], 3 Mandelbrot 8E M, ¦ λ ∈ H ⊂ M. -
Φ : M →M ÓN. K z0 = U ∩ Bλ PΦ−1(λ) β–ØÄ:[/Ó
. Ï β–ØÄ: Julia 8Ø, ù H 6= Φ(H♥) gñ. Úny.
½n 4.1.5 y². ·Äky²XJ λ÷v (a)½ (b),K Jλ ´ Sierpinski/
#. XJ Int(Kλ) = ∅,K Qλ = ∅. â½n 4.3.5,·Iy²ézØÓ V1, V2 ∈ Pk V 1 ∩ V 2 = ∅. ·Äk`² Bλ ∩ T λ = ∅. Ø,, aqu [24, ·K 4.3] y²,
±íÑ z0 ∈ ∂Bλ∩∂Tλ ´ fλ .:,ù (a)¥bgñ. b Vi f−kiλ (Bλ)
§4.4. CANTOR ±[é¡AÛ 69
©|,Ù¥ i = 1, 2 0 ≤ k1 ≤ k2. ·©ü«¹?Ø.XJ k1 = k2 V1 6= V2,
K V 1 ∩ V 2 = ∅. ÄK, aqu [24, ·K 4.3] y², ±íÑ ∂V1 ∩ ∂V2 c;¥
,:´ fλ .:, ù (a) ¥bgñ. XJ k1 < k2, K Bλ = f k2−1λ (V1)
Tλ = f k2−1λ (V2). d Bλ ∩ T λ = ∅ V 1 ∩ V 2 = ∅. Ïd, XJ Int(Kλ) = ∅ gd
.;4 ∂Bλ Ø, K Jλ ´ Sierpinski /#.
XJ λ ∈ H ∪ rH, Ù¥ H ´ M 3 Λ ¥ primitive V©|, ¿ H Ø´H♥ 3ÓNe.âÚn 4.3.3 Q¥ Fatou©|4pØ. d
, aquþã?Ø, P ¥ Fatou ©|4pØ. ·Iy² P¥ Fatou ©|4Ú Q ¥ Fatou ©|4pØ=.
- U ∈ Q á3 Kλ ¥ Fatou ©|. âÚn 4.3.6, U ∩Bλ = ∅, u´éz V ∈ P k U ∩ V = ∅. l fλ ¤k Fatou ©|4ÑpØ. Ïd, Jλ 3
ù«¹e´ Sierpinski /#.
XJ λ á3 Λ V©|¥, K Jλ ´ëÏ fλ .;4ØU
∂Bλ . Ï>. ∂Bλ ´[ ( [61, ½n 1.2]), |^½n 4.2.9 ¥
Ó?Ø, ±y² Jλ ÷v[Úé©l^. XJ rH ´ primitive V
©|, K frH Fatou 8¹kk:üëÏÔ Fatou ©|, ù©
|>.½Ø´[. Ïd JrH ØU[é¡du/#. y²..
§4.4 Cantor ±[é¡AÛ
lÿÀÝ5w,¤k Cantor±Ñ´,ù´Ï§ÑÿÀdu “I
O” Cantor ± C × S1, Ù¥ C n© Cantor 8 S1 ü ±. Ïd, ¤
k Cantor±¤8Ü´L(, ·±^[é¡AÛÝ5w. ¯¢þ,
[é¡AÛp¡Ä¯KÒ´äü½ÓÝþm´Ä´*d[é¡
d.
Ýþm X /ê confdim(X) ¤k X [é¡dÝþm Haus-
dorff êe(..
Ún 4.4.1. XJ n ≥ 3, K x = log(n)/ log(n+ 1) Ø´e§
l−x +m−x = 1, (4.4.1)
Ù¥ l,m ≥ 2 ü¦ 1/l + 1/m < 1 ê.
y². Ø5,·b 2 ≤ l ≤ m,K 1/lx ≥ 1/mx,Ù¥ x = log(n)/ log(n+1).
XJ n ≥ 3, du log(n− 1)/ log(n) < log(n)/ log(n+ 1), K
1
lx+
1
mx≤ 1
2log 3/ log 4+
1
3log 3/ log 4= 0.9960381127 · · · < 1.
70 1oÙ McMullen N Julia 8[é¡x
ùÒ¤Ún 4.4.1 y².
éz n ≥ 2, - Jp,d1,··· ,dn (3.1.2) ¥½Â fp,d1,··· ,dn Julia 8. e5, ·
o´b½ ai ®²½n 3.1.1 ¥@ÀЦ Jp,d1,··· ,dn ´ Cantor ±, Ï·
=éù«¹a,. Ó, ·b½ λ v¦ (4.1.1) ¥½Â McMullen N
fλ Julia 8 Jl,m Cantor ±, Ù¥ 1/l + 1/m < 1. XJéz 1 ≤ i ≤ n k
di = n+ 1, ·^ gn 5L« fp,n+1,··· ,n+1 - Jn ÙéA Julia 8.
½n 4.4.2. Jp,d1,··· ,dn /êu 1 + αd1,··· ,dn, Ù¥ αd1,··· ,dn e¡§
:n∑i=1
d−αd1,··· ,dni = 1.
AO/, XJéz 1 ≤ i ≤ nk di = n+1, K αn := αd1,··· ,dn = log(n)/ log(n+1). XJ
k 6= n, K αk 6= αn. XJ n ≥ 3, Kéz¦ 1/l + 1/m < 1 l,m ≥ 2 k αn 6= αl,m.
y². â½n 3.1.1 y², · fp,d1,··· ,dn |Ü3 Haıssinsky Ú Pilgrim
¿Âedêâ D := (d1, · · · , dn) ∈ Nn û½ [35, §2]. â [35] ¥·K 1.1 Ú 2.2,
fp,d1,··· ,dn Julia 8/ê´ confdim(Jp,d1,··· ,dn) = 1 + αd1,··· ,dn , Ù¥ αd1,··· ,dn ´
§∑n
i=1 d−αd1,··· ,dni = 1 . AO/, XJéz 1 ≤ i ≤ n k di = n + 1, K
αn := αd1,··· ,dn = log(n)/ log(n + 1). ùL² m 6= n du αm 6= αn. âÚn 4.4.1
, ½n¤á.
½n 4.4.2 ÑäN~f5y3ùVkn¼ê, Ù Julia 8
Cantor±/ê?¿Cu 2 ( [35,½n 2]). â½n 3.1.1y²,·
¤k fp,d1,··· ,dn Ñ´V. 5¿/ê´Ýþm[é¡ØCþ ( [47])
?¿Vkn¼ê Julia 8 Hausdorff êîu 2 ( [70, ½n 4 ÚíØ]).
Ïd, ½n 4.4.2 á=ke¡üíØ.
íØ 4.4.3. é?¿ k, n ≥ 2, Julia 8 Jk Ú Jn *d[é¡d= k = n.
d, XJ n ≥ 3, K Jn Ø[é¡du?Û Jl,m, Ù¥ 1/l + 1/m < 1.
íØ 4.4.4. Jn Hausdorff ê Hdim(Jn) ÷v
1 + log(n)/ log(n+ 1) ≤ Hdim(Jn) < 2.
íØ 4.4.3 Ú 4.4.4 y². XJ·5¿/ê´[é¡ØCþ, @o
ùüíØÑ´½n 4.4.2 íØ.
1ÊÙ x¼êÄåXÚ
éX¼ê Siegel >.ÿÀÚAÛ5ïÄ´~¯K. é
^=ê\þ½â^, Douady ß Siegel >.½´ Jordan . 3
ù¯Kþ, éõöÑLz. 8cÐ(Jáu Zhang Ú Zakeri. ^
=ê θ k.., ¦©Oy²NÝØ$u 2 kn¼ê±9äk/ª f(z) =
P (z) exp(Q(z)) 5¼ê Siegel >.Ñ´²L.:[, Ù¥ f(0) = 0
f ′(0) = e2πiθ, P Ú Q Ñ´õª.
3ùÙ¥, ·ò?رϼêx. §Ø¹3 Zakeri ¤Äx
¥. ·Óy², XJ^=êk.., @oT¼êx¥z±:% Siegel
>.½´²L.:[.
§5.1 cóÚy²g´Vã
- f 3:äkÃn¥5ØÄ:5X¼ê. =, f(0) = 0 f ′(0) =
e2πiθ, Ù¥ 0 < θ < 1 ´Ãnê. XJ3X¼êò f Ýf5^=
Rθ : z 7→ e2πiθz, K¡ f 3ØÄ: 0 ?´ÛÜ5z. ¼ê f Ýu Rθ «
´üëÏ« ∆f , ¡´ f ±:% Siegel . Ï f 3 Siegel ∆f
p¡ÄåXÚ´ü. éõÿ, ·'%>. ∂∆f AÛÚÿÀ5. ~X, §´
Ä´^ Jordan ? ½ö?Ú, §´Ä´[?
éuX¼ê, ù¯Kvk)û. 3½^, éõöé
X¼êx)ûù¯K.AO´3^=ê θk..¹e,kéõ'(J.
5¿XJÃnê 0 < θ < 1멪Ðm θ = [a1, a2, · · · , an, · · · ]÷v supan <∞,
K¡ θ ´k... 3ù^e, Douady, Zakeri Ú Shishikura, ©Oy²gõ
ª, ngõª±9¤kNÝØu 2 õªk.. Siegel >.´[
[26, 77, 66]. éõöé¼ê/Äù¯K. Geyer, Keen Ú Zhang ©
OÄ z 7→ e2πiθzez Ú z 7→ (e2πiθz +αz2)ez [32, 41]. Ó, Cheritat Ä“
ü”¼ê [14].
C, Zakeri Ú Zhang ©O(J. éu5¼ê f(z) =
P (z) exp(Q(z)), Ù¥ f(0) = 0 f ′(0) = e2πiθ, P Ú Q Ñ´õª, Zakeri y²XJ
0 < θ < 1 ´k..Ãnê, @o f ±:% Siegel >.´[
¹ f .: [78]. 3 [80] ¥, Zhang y²ù(Øé¤kNÝØ$
u 2 kn¼ê¤á.
Zhang ÓéÙ§±Ï¼êÄù¯K. ¦e¡(J.
71
72 1ÊÙ x¼êÄåXÚ
½n (Zhang, [79]). - 0 < θ < 1 k..Ãnê. K¼ê f(z) =
e2πiθ sin(z) Siegel >.´TвLü.: π/2 Ú −π/2 [.
3ùÙ¥, ·ò?رϼêx. - 0 < θ < 1 k..Ãn
ê. P
F = fα(z) = e2πiθ sin(z) + α sin3(z) : α ∈ C \ 0. (5.1.1)
éz fα ∈ F , ·y²e¡
½n 5.1.1. ¼ê fα ±:% Siegel >.´[, 3Pê¿
Âe²L fα 2 , 4 ½ö 6 .:.
½n 5.1.1 y²´É Keen Ú Zhang [41, 79] óéu. ·|^[/N
E|5Ex¼êx,ùx¼êx¥z¼êÑäk±:% Siegel,
ù Siegel ´dx3:äkAÛá5ØÄ:¼êxÏLé5z«Ã
â=z5. ùpEI±±Ï5, aq/EkÑy3 [79] ¥.
·`²e½n 5.1.1 y² Keen-Zhang [41] y²ØÓ?. ¦Ä
¼êx¥z¼ê=küIP.:, ¤±§éAëêmkg,y
©. äN/, §ÄkÚ\x Σ′1/2 = gα(z) = (z/2 + αz2)ez : α ∈ C \ 0 ,é^ Jordan C ⊂ Σ′1/2 ò Σ′1/2 ©¤üÜ© Ai, Ù¥ i = 1, 2. ,¦`²XJ
α ∈ Ai, K±:%5z«>. ∂Dα TÐ=²L gα .: ciα,
Ù¥ i = 1, 2. XJ α ∈ C, K ∂Dα ²Lü.:. ·Ä¼êxkáõ
.:. ¦+ù¼êÑ´±Ï, ´éJ(½=.:±á3±:%
5z«>.þ. , , |^, “f5” ?Ø, ·±ëêm
aqy© (½n 5.1.2 Ú §5.3).
·ÄkVãe½n 5.1.1 y²g. 3ùÙÜ©, ·½k.
.Ãnê 0 < θ < 1 - λ = e2πiθ. N´/IP.:, ·½ÂBf
Λ = c : |Re(c)| ≤ π/2 \ 0. (5.1.2)
éz½ t 6= 0, Äe¡ëêm:
Σt =
fc(z) = t
(sin(z)− sin3(z)
3 sin2(c)
): c ∈ Λ
. (5.1.3)
OL²éz fc ∈ Σt k f ′c(z) = t sin(z − π/2) sin(z + c) sin(z − c)/ sin2(c). l
fc .:8 Critc = π/2 + kπ, ±c + kπ : k ∈ Z. AO/, XJ c = π/2 ½ö
−π/2, K fc äkê 3 .: π/2 + kπ, Ù¥ k ∈ Z. Ï fc ´±Ï 2π
§5.1. cóÚy²g´Vã 73
Û¼ê, XJ c 6= π/2 Ú −π/2, K fc þkgd.:. Ï éu fc, ·
IÄ.: c Ú π/2 ÄåXÚ5=.
,, ·ÄüAÏëêm Σ1/2 Ú Σλ. ëêm Σ1/2 d@3:äk
¦f 1/2 á5ØÄ:¼ê¤, Σλ K¹½n 5.1.1 ¥¤k¼ê (Ún
5.2.3). ùp, Σ1/2 ¥eI 1/2 Ø´, §±d?Û÷v 0 < |t| < 1 t O.
éz fc ∈ Σ1/2, - Ωc fc ±:%5z«.·òy² Ωc ´
k.üëÏ«. éu Λ ¥^©l 0 Ú ∞ ü4 γ, ·^ γint Ú γext
5©OL« Λ \ γ k.ÚÃ.©| (5¿ Λ \ γ Ã.©|UØ). ÏLïÄ
ëêm Σ1/2 ¥¼êÄåXÚ5, ·kXe©a½n.
½n 5.1.2. 3²L π/2 Ú −π/2 ©l 0 Ú ∞ ü4 γ ⊂ Σ1/2 ¦
(1) XJ c ∈ γint, K ∂Ωc TвL 2 .: c Ú −c.(2) XJ c ∈ γext, K ∂Ωc TвL 2 .: π/2 Ú −π/2.
(3) XJ c ∈ γ \ ±π/2, K ∂Ωc TвL 4 .: ±π/2 Ú ±c.(4) XJ c = π/2 ½ −π/2, K ∂Ωc ²L 2 ê 3 .: π/2 Ú −π/2.
Ø5, éu c ∈ γext ∪ γ, ·^ fc ∈ Σ1/2 5E.N Fc. E
L§Ì´ò5z« Ωc =z¤ Fc ^=. ,, é Fc–ØC
Beltrami Xê νc. ÏL)äkXê νc Beltrami §, ·± 0, 2π Ú ∞ØÄ[/N wc. Ïd Tc = wc Fc w−1
c 3 C þ´)Û. ÏL?Ø Tc ":©
Ù±9Ù§&E, ±y² Tc ∈ Σλ (Ún 5.2.5 ÚíØ 5.5.9).
ÏLù«ª, ·ÃâN S : γext ∪ γ → Σλ, ½Â S(fc) = Tc. ,,
S ±aq/l γint ∪ γ Σλ ½Â. ,, ·y² S ´ëY S(γ) k½n
5.1.2 ¥ãaq5.
!, ·y²ÃâN S : Σ1/2 → Σλ ´÷. ùÄuy² S : γint ∪ γ →S(γ)int ∪ S(γ) Ú S : γext ∪ γ → S(γ)ext ∪ S(γ) Ñ´÷ (·K 5.6.4). ù·B
½n 5.1.1 ÚXe½n.
½n 5.1.3. - F = fα : α ∈ C \ 0 (5.1.1) ¥½Â¼êx. K3ü
H : C \ 0 → Λ Ú^²L −λ/3 ©l 0 Ú ∞ ü4 Γ ⊂ C \ 0 ¦(1) XJ α ∈ Γint, K ∂∆fα TвL 2 .: π/2 Ú −π/2.
(2) XJ α ∈ Γext, K ∂∆fα TвL 2 .: H(α) Ú −H(α).
(3) XJ α ∈ Γ \ −λ/3, K ∂∆fα TвL 4 .: ±π/2 Ú ±H(α).
(4) XJ α = −λ/3, K ∂∆fα TвL 2 ê 3 .: π/2 Ú −π/2.
½n 5.1.3 Ñaqu Zakeri ng Siegel õª [77] ±9 Keen-Zhang ¼ê
x [41] ëêmaqy©.
74 1ÊÙ x¼êÄåXÚ
§5.2 Ä(J
- f : R→ R ¢¶Ó. XJ3~ê k ¦
1
k≤ f(x+ t)− f(x)
f(x)− f(x− t)≤ k (5.2.1)
é¤k x ∈ R Ú t > 0 ¤á, KN f ¡´[é¡. ~ê k ¡´[é¡~ê.
â Beurling Ú Ahlfors ½n [7], ^ (5.2.1) du f kl R þ²¡H [/*Ü. AO/, ¦y²½Â
fBA(x+ iy) =1 + i
2
(∫ 1
0
f(x+ ty)dt− i∫ 1
0
f(x− ty)dt
)(5.2.2)
N fBA : H→ H´ f K–[/*Ü,Ù¥K ≤ k2. ù*Ü¡ Beurling-
Ahlfors *Ü.
- f : S1 → S1 Ó F : R → R f 3 τ(x) = e2πix eJ,, Ù¥
τ ´l R S1 CXN. 5¿ F ÷v F (x + 1) = F (x) + 1 3\
~ê¿Âe´. XJé F k (5.2.1) ¤á, KN f : S1 → S1 ¡´ k-[é¡
. u´, F k Beurling-Ahlfors *Ü FBA, ù*Ü´l H ½ÂgC. 3CXN
τ(z) = e2πiz e, du FBA(z + 1) = FBA(z) + 1, FBA g,pl D NgC[/N. ·^ fBA L«ùN, §´[/N±:ØÄ. Ó, N
fBA : D→ D ¡´ f : S1 → S1 Beurling-Ahlfors *Ü.
- γ ^ü4, XJ3[/N fγ : C→ C ¦ γ = fγ(S1), K
γ ¡´[. AO/, XJ fγ ûØL K, K γ ¡´ K–[.
- f : C → C, g : S1 → S1 ±9 γ ©O[/N, [é¡NÚ
[. ·^ K(f), K(g) Ú K(γ) 5©OL« f , gBA ±9 fγ û.
^ (½4l) ¡´¢)ÛXJ§´± (½ö4ã) 3½Â3T
± (½ö4ã) NC/N. ^ml¡´¢)ÛXJ§¤k4flÑ
´¢)Û ( [43, p.20]). e¡Ún´ Schwarz n/ª ( [1]).
Ún 5.2.1. D «, γ ¹3 ∂D ¥m¢)Ûã. b f ´
½Â3 D þX¼ê, ¦ f ±ëY/òÿ γ þ, f(γ) ´¢)Û
ã. @o, f ±X/òÿ¹ D «þ, ¦ γ ¹3T«
SÜ.
b f : S1 → S1 ´Ó f 3 S1 þ^=ê´Ãnê θ. XJ3
½Â3ü ±þ[é¡N ϕ ¦ ϕ f ϕ−1 = Rθ 3 S1 ¤á, KN f ¡
§5.2. Ä(J 75
´±[é¡5z, Ù¥ Rθ L«f5^= z 7→ e2πiθz. 5¿XJ\þ5z^
ϕ(1) = 1, K ϕ ´.
e¡[é¡5z½nw·¢)Û±ÓÛ±[é¡5z.
½n 5.2.2 (Herman-Swiatek, [37, 72]). - f : S1 → S1 ^=ê θ ¢)Û
.±Ó. K f [é¡Ýu Rθ = θ k... d, [é¡~ê=6u
θ Ú f ±)Ûòÿ¹ü ±.
éz t ∈ C\0,- ΛÚ Σt (5.1.2)Ú (5.1.3)¥½Â8Ü.5¿ c 7→ sin2(c)
´±Ï π ó¼ê. ·k
Ún 5.2.3. ¼ê τ(c) = −λ/(3 sin2(c)) ´l Λ C \ 0 ÷.
ùÚnL² Σλ ¹½Â3 (5.1.1) ¥ F ¥¤k. |^ü?
Ø, N´wÑ τ : Λ→ C \ 0 “Øõ” ´NÝ 2 XCX.
ëêm Σt ¥¼êäkkõ.vkkì? (Ún 5.3.2). Ï
§´±Ï, § Fatou ©|©a~ü. ~X, ®²§vkiÚ
Baker [33]. § Fatou8õkü;. XJ t = 1/2½ö λ, 3 Fatou
©|;ªÑS¹: Fatou ©|þ, ù Fatou ©|©OéAáÚ
½ Siegel . d, U3Ù§ Fatou ©|;, ù¿Ø.
½Â 5.2.4. f : C→ C ´¼ê, K f O ρ(f) ½Â
ρ(f) = lim supr→+∞
log logM(f, r)
log r, (5.2.3)
Ù¥ M(f, r) := sup|z|=r |f(z)|. XJ ρ(f) < +∞, K¼ê f ¡´k.
N´y, ~X, ëêm Σt ¥z¼êÑ´k, Ù¥ t 6= 0.
Ún 5.2.5. b f : C → C ´k¼ê, Ó´±Ï 2π
Û¼ê. d, f ′(0) = t 6= 0 f ":8´ kπ,±a + kπ : k ∈ Z, Ù¥ a 6= 0. K
f ∈ Σt.
y². âÚn 5.2.3, 3 c ∈ Λ, ¦√
3 sin(c) = sin(a). éu fc ∈ Σt, fc Ú f
":8´. K f/fc Û:´3 C þk f/fc 6= 0. ùL² f = fc exp(h),
Ù¥ h ´,¼ê. 5¿ f Ú fc Ñ´Û¼ê, ùL² h Ó´ó¼ê.
Ï fc k, ùL² exp(h) ´Xd. u´ h 7õª. Ï f ±
Ïu 2π ¼ê, K3ê k ¦ h(z + 2π) = h(z) + 2kπi éz z ¤á.
¯¢þ, Ï h ´ z ëY¼ê, ù k ´½e5. P H(z) = h(z) − ikz, ·k
76 1ÊÙ x¼êÄåXÚ
H(z + 2π) = H(z). u´ H ´±Ïõª, l 7~ê. ùL² h(z) = ikz + b
é,~ê b ¤á. Ï h ´óê, ¤± k 7u 0. 5¿ f ′(0) = t, Ïd eb = 1. Ú
ny²..
§5.3 5z«
3Øu· ¹e, ·ò Σt ¥¼ê fc Ú Λ ¥ëê c w¤´. é
z c ∈ Σt, fc Q´Û¼êq´±Ï¼ê. Ïd z Ú −z 3 fc eÄåXÚ1
3==ÎÒ ± ¿Âe´.
3ù«¹e, · fc Fatou 8Ú Julia 8'u:é¡. AÏ:, '
X, .:7L¤éÑy.
Ún 5.3.1. ¼ê fc Fatou 8'u:é¡äk±Ï π.
y². 5¿ fc ´Û¼êäk±Ï 2π. éz n ≥ 1 Ú k ∈ Z, XJ k Ûê,
K f nc (z + kπ) = −f nc (z) = f nc (−z). XJ k óê, K f nc (z + kπ) = f nc (z).
â5x½Â,· f nc n≥1 3: z,−z Ú z+ kπ ?Ó5½ÓØ
5. y²..
XJ3 C ¥^ëY γ(t) : [0,∞) → C ÷v limt→∞ γ(t) = ∞, ¿¦
limt→∞ f(γ(t)) = ζ, K: ζ ¡´¼ê f ìC.
Ún 5.3.2. éz c ∈ Σt, fc vk?ÛkìC.
y². ·^yy². b ζ ´ fc kìC.â½Â,3^ë
Y γ(t) = x(t) + y(t)i : [0,∞)→ C, ¦ γ(t)→∞ t→∞ k fc(γ(t))→ ζ.
Ï fc ´±Ï 2π ëY¼ê, K3 0 < t1 < t2 < · · · < tn < · · ·¦ tn → ∞ n → ∞ k y(tn) → ∞. OL² y(tn) → ∞ kfc(γ(tn))→∞. ù´gñ.
éz c ∈ Σ1/2, fc 3:ká5ØÄ:. â Kœnigs 5z½n, 3½Â
3 0 SX¼ê φc, ò fc ÝØ N L1/2(w) = w/2. ù5z¼ê φc
¡´ Kœnigs ¼ê. â [53, íØ 8.4], 3:áÚþf,
¦3ùfþ, fc XÝu L1/2. ùf¡´ fc ±:%5z
«. ·^ Ωc L«. XJ\þ5^ φ′c(0) > 0, K5zN φc : Ωc → D ´. d, φc ±)Û/òÿ Ωc ;þ.
íØ 5.3.3. 5z« Ωc ´k.'u:é¡. d, φc ´Û¼ê.
§5.3. 5z« 77
y². XJ Ωc ´Ã., K fc(∂Ωc) ¹ fc kìC, ùÚn 5.3.2
gñ. é¡5âÚn 5.3.1 .
éu1(Ø, 5¿ −φc(−z) 3 Ωc þÓò fc Ý L1/2, −φc(−z) 3
:äkê. â5, ùL² φc ´Û¼ê.
Ún 5.3.4. éu¤k c ∈ Σ1/2, >. ∂Ωc ´'u:顲Lüé
¡.:[.
y². â [53] ¥Ún 8.5 y²L§, · ∂Ωc ¹ fc .:.
âÚn 5.3.1 ¥¤y²é¡5, ∂Ωc 'u:顲Lüé¡.:.
XJ·£ ∂Ωc þ¤k.:, Kekõ^ml, §Ñ´¢)Û. d
, §3.:?±"Ý. ùL² ∂Ωc ´[ ( [43, p.104]).
âÚn 5.3.4, éu?¿ c ∈ Σ1/2, K73,.:á3 ∂Ωc þ. ·
'%§´=.:. 5¿ fc .:¤8Ü´ Critc = π/2 + kπ, ±c + kπ :
k ∈ Z. ù!eÜ©òy²e¡½n.
½n 5.3.5 ((½n). 3²L π/2 Ú −π/2 ü γ ⊂ Σ1/2 ©l 0 Ú ∞,
¦XJ c á3 γ SÜ, K ∂Ωc TвLü.: ±c; XJ c á3 γ Ü, K
∂Ωc TвLü.: ±π/2.
¯¢þ, ·Äk±y² cá3:S, ∂Ωc =²Lü.
: cÚ −c. éu Im(c)é/, ·±y² ∂Ωc =¹ü.: π/2Ú −π/2.
ù´Ï3l:é/, fc Ú sin(z)/2 ~C.
éz k ∈ Z, P Ωkc = z + kπ : z ∈ Ωc. 5¿ Ω0
c = Ωc.
Ún 5.3.6. XJ i 6= j, K Ωic ∩ Ωj
c = ∅, Ù¥ i, j ∈ Z. d, éu i 6= j, XJ
Ωi
c ∩ Ωj
c 6= ∅, K3, k ∈ Z ¦ Ωi
c ∩ Ωj
c = π/2 + kπ.
y². Ø5, béu, k0 6= 0 k Ωc ∩ Ωk0c 6= ∅. é?¿áu Ωc ∩ Ωk0
c
z 6= 0, · ±z ±(z− k0π)Ñá3 Ωc ¥. Ïé?¿ k ∈ Zk π/2 + kπ 6∈ Ωc,
ùL² z 6= k0π − z. XJ k0 Ûê, K fc(z) = fc(k0π − z); XJ k0 ´óê, K
fc(z) = fc(z − k0π). Ï fc 3 Ωc þ´ü, ù´gñ. Ï fc 3 Ωc þÓ
´üéu, k0 ∈ Z, z = k0π − z ∈ Ωc ´#N, ùL²1(ؤá.
íØ 5.3.7. é?¿ c ∈ Σ1/2, 3, k ∈ Z ¦ ∂Ωc ∩Critc ⊂ π/2,−π/2, c+
kπ,−c− kπ.
78 1ÊÙ x¼êÄåXÚ
y². XJ3 c ∈ Σ1/2 ÚüØÓ k1, k2 ∈ Z, ¦ ∂Ωc ¹ 4 ØÓ.
: ±(c+ k1π) Ú ±(c+ k2π), K Ωc ∩Ωk1−k2c ¹ c+ k1π Ú −(c+ k2π), ùÚn 5.3.6
gñ. ùL² ∂Ωc õ¹é/X ±(c + kπ) .:, Ù¥ k ∈ Z. aq/, ∂Ωc
¹õéäk/ª ±(π/2 + kπ) .:, Ù¥ k ∈ Z.
b3 c ∈ Σ1/2 ¦ ±(π/2 + k0π) ∈ ∂Ωc é, k0 ≥ 1 ¤á. 5¿ φc : Ωc → D\5z^ φ′c(0) > 05zN. - I := [φc(π/2 +k0π),−φc(π/2 +k0π)]
D ¥ë φc(π/2 + k0π) Ú −φc(π/2 + k0π) », K J0 := φ−1c (I) ´^ë
π/2 + k0π Ú −(π/2 + k0π) 'u:é¡ Jordan . P Jn = z + (2k0 + 1)nπ :
z ∈ J0, K J := ∪n∈ZJn ´±Ï (2k0 + 1)π ±Ï. é?¿ x ∈ R, P
Jx = z + x : z ∈ J, K J ∩ Jx 6= ∅. AO/, J ∩ Jπ 6= ∅. â±Ï5Úé¡5, 3,
z ∈ J ∩ Jπ ¦ Re(z) ∈ [0, π/2 + k0π]. d, Ï J0 ∈ Ωc, z ∈ Ωc ∩ Ω1c . âÚ
n 5.3.6 z = π/2 + k0π. ¯¢þ, XJ3 k 6= k0 ¦ z = π/2 + kπ ∈ J ∩ Jπ, K
π/2 + kπ ∈ Ωc ´ØU. y3· ∂Ωc ¹4 ØÓ.: ±(π/2 + k0π)
±(π/2− k0π). ùþã?Øgñ.
5¿íØ 5.3.7 ¥ k U6 c. y²½n 5.3.5, ·I`²éz c Ñ
k k = 0.
- fn : C → C X¼ê, Ù¥ fn(0) = 0, n = 1, 2, · · · . b n ªu
∞ , fn ÛÜÂñ f∞ 0 < |f ′∞(0)| < 1, K f∞ ´±:ØÄ~
X¼ê. ùL² f∞ 3:kAÛá5ØÄ:. u´éu¿© n, ·k
0 < |f ′n(0)| < 1. Ø5, béu¤k n = 1, 2, · · · , fn k±:%5z« Ωn. ?Ú, b f∞ ±:%5z« Ω∞ ´k..
6/, · Kœnigs ¼ê φn : Ωn → C 5z^ φ′n(0) = 1. ¯¢þ,
φn ½Â φn(z) = limk→∞ fkn (z)/(f ′n(0))k, Ù¥ 1 ≤ n ≤ ∞. 3¢ê rn > 0
¦ φn : Ωn → Drn ´/ limn→∞ rn = r∞. ½Â/N Gn : Ω∞ → Ωn
Gn(z) = φ−1n ( rn
r∞φ∞(z)), §±ëY/òÿ ∂Ω∞ þ. - An fn ¹ 0 áÚ
. Ï fn ÛÜ/Âñ f∞, ùL²XJ;8 E∞ ¹3 A∞ S, Kéu¿©
n, k E∞ ¹3 An S. AO/, XJ n ¿©, K3 Ω∞ m Ω′∞, ¦
Ω∞ ⊂⊂ Ω′∞ φn 3 Ω′∞ þ½Â´Ün.
·K 5.3.8. n ªu ∞ , 5z« Ωn Ú§>. ∂Ωn 3 Hausdorff
ÿÀ¿Âe©OÂñ Ω∞ Ú ∂Ω∞.
y². ·Äky² Gn : f∞(Ω∞) → fn(Ωn) ÛÜÂñðN. Ï
φ∞(f∞(Ω∞)) = Dρr∞ φn3 Ω′∞þÛÜÂñ φ∞,3 ε > 0¦ φn(f∞(Ω∞)) ⊂
§5.3. 5z« 79
Dρr∞+ε ⊂ D(ρ+1)rn/2, Ù¥ ρ = |f ′∞(0)| ∈ (0, 1). ùL² φ−1n ±3 φn(f∞(Ω∞)) þ½Â.
éu z ∈ f∞(Ω∞), 3 ξn ∈ φn(f∞(Ω∞)) ¦
|Gn(z)− z| = |Gn(z)− φ−1n φn(z)| ≤ |(φ−1
n )′(ξn)| |rnφ∞(z)/r∞ − φn(z)|. (5.3.1)
5¿ f∞(Ω∞) ´;, f∞(Ω∞) ⊂⊂ Ω∞ φn 3 Ω′∞ þÛÜÂñ φ∞, K3~
ê a > 0 ¦éu¿© n k |φ′n(z)| ≥ a 3 f∞(Ω∞) ¤á. â (5.3.1), ùL²
n ªu ∞ |Gn(z) − z| ≤ 1ar∞
(rn|φ∞(z) − φn(z)| + |φn(z)||rn − r∞|) → 0, ù´Ï
(φ−1n )′(ξn) = 1/φ′n(ξ′n) é, ξ′n ∈ f∞(Ω∞) ¤á. l Gn : f∞(Ω∞)→ fn(Ωn) ÛÜ
ÂñðN.
é?¿ 0 < r < r∞, |^þãaq?ر`² Gn : φ−1∞ (Dr) →
φ−1n (Drrn/r∞) ÛÜÂñðN. Kéu?¿ ε > 0, 3¿© N > 0 ¦
n ≥ N , φ−1∞ (Dr∞−2ε) ⊂ Gn(φ−1
∞ (Dr∞−ε)) ⊂ Ω∞. â Gn ½Â, ù¿X
φ−1∞ (Dr∞−2ε) ⊂ Ωn.
À¿© ε > 0 ¦ Ω∞ ⊂⊂ Ωε∞ ⊂ Ω′∞, Ù¥ Ωε
∞ := z : d(z,Ω∞) < ε. Ï Ω∞ ´k., ùL² Ωε
∞ ´;. du φ∞(Ω∞) = Dr∞ , l Dr∞ ⊂⊂ φ∞(Ωε∞). -
δ(ε) > 0 ∂Dr∞ Ú ∂φ∞(Ωε∞) mî¼ål. Ï limn→∞ rn = r∞, 3 N1 > 0, ¦
n ≥ N1 , k rn < r∞ + δ(ε)/2. Ï φn 3 Ωε∞ Ú ∂Ωε
∞ þÛÜÂñ φ∞,
ùL²XJ n ¿©, K Drn ⊂⊂ φn(Ωε∞).
du φn(0) = 0 Ú φ−1∞ (Dr∞−2ε) ⊂ Ωn, 3 r > 0 ¦ φ−1
n ±l Dr
½Â φ−1n (Dr) ⊂ Ωn ∩ Ω∞. Ï Drn ع φn ., u´ φ−1
n 3 Dr þ
±)Û/òÿ φ−1n |Drn . ¯¢þ, Ï φ−1
n (Dr) ⊂ Ωn, φ−1n (Drn) = Ωn. du
Drn ⊂⊂ φn(Ωε∞), ùL² Ωn ⊂ Ωε
∞. y²..
éu?Û'ê R > 0, P ΛR = c ∈ Σ1/2 : |c| ≥ R. éu?Û'êε > 0, P Λε = c ∈ Σ1/2 : 0 < |c| ≤ ε.
Ún 5.3.9. 3 R > 0 ¦éuz c ∈ ΛR, ∂Ωc TвLü.: π/2 Ú
−π/2. Ó, 3 ε > 0 ¦éz c ∈ Λε, ∂Ωc TвLü.: c Ú −c.
y². Ï c3 Σ1/2¥ªu∞, fcÛÜÂñ f∞(z) = sin(z)/2,u´
â·K 5.3.8 Ωc3 HausdorffÿÀeÂñ Ω∞. ùp, Ω∞L« f∞5z«
. §´k., 'u:顲L π/2 Ú −π/2 ( [79, Ún 2]). Ï 3 Ω∞
k. Ω′∞, ¦ Im(c) ¿©, k Ωc ⊂⊂ Ω′∞ Ω′∞ ∩Critc = π/2,−π/2.âÚn 5.3.4, Im(c) ¿©, ùL² ∂Ωc TвLü.: π/2 Ú −π/2.
éu c á3 0 /, 5¿ c ªu 0 , L−1c fc Lc(z) ÛÜ
/Âñngõª f0(z) = z/2 − z3/6, Ù¥ Lc(z) = cz. ùL²3 Hausdorff ÿÀe,
80 1ÊÙ x¼êÄåXÚ
L−1c fc Lc(z) 5z« Ω′c ÛÜÂñ f0 5z« Ω0. §´
k.Ï f0 ´õª. u´3 Ω0 k. Ω′0, ¦ c v
ÿ, k Ω′c ⊂⊂ Ω′0, ùdu cΩ′c ⊂⊂ cΩ′0, Ù¥ cΩ′c := cz : z ∈ Ω′c = Ωc. ùL² c ¿
©ÿ, ∂Ωc ==¹.: c Ú −c. y²..
5¿ Λ = c : |Re(c)| ≤ π/2 \ 0 þ!¥½Âëêm, u´ sin : Int(Λ) →C\(−∞,−1)∪0∪(1,+∞)´/. éz β ∈ (0, π)∪(π, 2π),- ηβ(t) = sin−1(teiβ :
0 < t < +∞). K ηβ ´ Σ1/2¥ë 0Ú∞÷v limt→0 ηβ(t) = 0Ú limt→∞ ηβ(t) =∞^1w.
âÚn 5.3.9, 3 Σ1/2 ©l 0 Ú ∞ ;f8 M , ¦ c á3 Σ1/2 \M k.©| (½Ã.©|) , K ∂Ωc TвLü.: c Ú −c (½ π/2 Ú −π/2). é
z β ∈ (0, π) ∪ (π, 2π), P
tβ = sup t : éu 0 < s < t, ∂Ωηβ(s) TвLü.: ± c. (5.3.2)
â tβ ½Â, ·±é sk → t−β ¦éz k k ±c ∈ ∂Ωηβ(sk). â·K
5.3.8, Ï ∂Ωηβ(t) 3 Hausdorff ÿÀ¥ëY6u t, ùL²48 ∂Ωηβ(tβ) Ó¹ ±c.â tβ ½ÂÚíØ 5.3.7, · ∂Ωηβ(tβ) Ó¹ ±π/2.
- γ ±π/2 Ú ηβ(tβ) : β ∈ (0, π) ∪ (π, 2π) ¿, ·k
Ún 5.3.10. éz c ∈ γ \ ±π/2, >. ∂Ωc TвL 4 .: ±π/2 Ú ±c.d, γ ´^ Jordan .
y². ·Iy²1(Ø.½Â χ : (0, π)∪ (π, 2π)→ γ \±π/2 χ(β) =
ηβ(tβ). â ηβ(t) ëY5Ú·K 5.3.8 , é?¿ β0 ∈ (0, π) ∪ (π, 2π) Ú
βn → β0, XJ n ¿©, K |ηβn(tβn) − ηβn(tβ0)| Ú |ηβn(tβ0) − ηβ0(tβ0)| Ñ¿©. ùL
² |χ(βn)− χ(β0)| v χ 3 β0 ?´ëY.
½Â χ(0) = π/2 χ(π) = −π/2. - εn > 0 Âñ 0 S, K c :
3 cn ∈ ηεn ¦ limn→∞ cn = c 3 Σ1/2 ¥à:´ E = (0, π/2) ∪ π/2 + iy :
y ≥ 0. ·äóXJ 0 < c < π/2, K ∂Ωc TвLü.: c Ú −c, XJc ∈ π/2 + iy : y ≥ 0, K ∂Ωc TвLü.: π/2 Ú −π/2. ¯¢þ, XJ c ∈ E,
K sin2(c) ∈ R, ùL² Ωc Ó'u¢¶é¡. XJ 0 < c < π/2, Ï Ωc ´üëÏ,
K ∂Ωc ∩ Critc = c,−c. XJ c ∈ π/2 + iy : y ≥ 0, KâÚn 5.3.6 1(Ø
k ∂Ωc ∩ Critc = π/2,−π/2. â tβ ½Â, N´wÑ limn→∞ ηεn(tεn) = π/2. XJ
ε′n < 2π ´Âñ 2π S, K|^aq?Øk limn→∞ ηε′n(tε′n) = π/2. ùL²
χ 3 0 ?´ëY. Ó/, ±`² χ 3 π ù:´ëY. y3·këY
V χ : [0, 2π)→ γ, du χ(0) = χ(2π−) = π/2, ùL² γ ´^ Jordan .
§5.3. 5z« 81
e5ò¬y² γ TÐÒ´½n 5.3.5 ¥ãü4. éu?Û½äk
JÜ c ∈ γ, ·Ñ fc |Ü&E. äkKJÜ c ∈ γ ±aq/©Û.
Ø3 ±π/2 Ú ±c ù 4 :, >. ∂Ωc ´¢)Û. 3 f−1c (fc(Ωc)) 4 k
.üëÏ©|N3 Ωc þ, Ù¥ Ω−1c Ú Ω1
c ©ON3 −π/2 Ú π/2 þ, Ù§ü
©| D1 Ú D2 ©ON3 c Ú −c þ. u´ f−1c (fc(Ωc)) =
⋃k∈Z Ωk
c ∪ Dk1 ∪ Dk
2 , Ù¥
Dki := z + kπ : z ∈ Di i = 1, 2. N´wÑ f−1
c (Ωc \ fc(Ωc)) düüëÏ©| E+
Ú E− ¤, §©O uëÏ8 f−1c (fc(Ωc)) þ¡Úe¡, §wå5´ü^Ã
f. ã 5.1.
π/2−π/2
c
−c
ΩcΩ−1c Ω1
c
D1
D2
D−11
D12
E+
E−
ã 5.1 fc ∈ Σ1/2 ÄåXÚ²¡, Ù¥ c = 0.944180 . . . (1 + i) ¦ ∂Ωc TÐ
²Lüéé¡.: ±π/2Ú ±c. ¥mxÚÜ©á3 fc Fatou8¥,
ÚÜ©KL«3 fc Se<º∞@:. Fatou8¥L«@
¦ |φc(z)| = |φc(c)|/2n :, Ù¥ n ∈ Z. ã: [−4.0, 4.0]× [−2.7, 2.7].
5¿ φc : Ωc → D \5^ φ′c(0) > 0 5zN. éu?¿ 0 ≤ r ≤ 1,
½Â ηr(t) := φ−1c (reit), Ù¥ t ∈ R. 3 t0 ∈ [0, 2π) ¦ η1(t0) = c. AO/,
η1(t0 + 2kπ) = c Ú η1(t0 + (2k+ 1)π) = −c, Ù¥ k ∈ Z. éu?¿ r ∈ (1/2, 1], - C+r
Ú C−r ©O ηr 3 E+ Ú E− ¥J,.äN/`,éu t ∈ Rk fc(C±r (t)) = ηr(t).
?Ú,éu½ t,·¦J, C+r (t)Ú C−r (t)ëY6u r ∈ (1/2, 1].
éu t ∈ R, P C+1/2(t) = limr→(1/2)+ C
+r (t) C−1/2(t) = limr→(1/2)+ C
−r (t). N´wÑ
C+1/2 Ú C−1/2 Ñ´ f−1
c (fc(∂Ωc)) = C+1/2 ∪ C
−1/2. - c = C+
1/2(t0) = C+1/2(t0 + 2π)
−c = C−1/2(t0−π) = C−1/2(t0+π). éu?¿ z ∈ E+½ö E−,3 r ∈ (1/2, 1]Ú
t ∈ R¦ z = C+r (t)½ö C−r (t). AO/,éu k ∈ Z,·k c+kπ = C+
1/2(t0−3kπ) =
82 1ÊÙ x¼êÄåXÚ
C+1/2(t0 + 2π − 3kπ) −c+ kπ = C−1/2(t0 − π + 3kπ) = C−1/2(t0 + π + 3kπ).
3 t1 ¦ −π/2 = η1(t1) 0 < t1 − t0 < π. ùL² π/2 = η1(t1 + π). é
u k ∈ Z, P Hk = z : (k − 1)π ≤ Re(z) ≤ (k + 1)π/2πZ, K fc : Hk → C ´NÝu 6 ©|CX. AO/, fc : Hk ∩E+ (½ Hk ∩E−)→ Ωc \ fc(Ωc) ´NÝ
u 3 ©|CX. éu c, −π/2, −c Ú π/2, §3 C+1 ∩Hk Ú C−1 ∩Hk þ©Ok 3
c. ùc±^S. äN/, c, −π/2, −c Ú π/2 3 C±1 þc´ C±1 (t0 + 2kπ),
C±1 (t1 + 2kπ), C±1 (t0 + π + 2kπ) Ú C±1 (t1 + π + 2kπ), Ù¥ k ∈ Z.
P D3 =⋃k<0 Ωk
c ∪ Dk1 ∪ Dk
2 , D4 =⋃k>0 Ωk
c ∪ Dk1 ∪ Dk
2 D =⋃
1≤i≤4Di. ·k
f−1c (D) =
⋃1≤i≤4,k1∈Z(D+
i,k1∪D−i,k1), Ù¥ D±1,k1 , D
±2,k1
, D±3,k1 Ú D±4,k1 ©O f−1c (D) N
3 C±1 (t0 + 2k1π), C±1 (t0 +π+ 2k1π), C±1 (t1 + 2k1π)Ú C±1 (t1 +π+ 2k1π)þ©|. £
E± =⋃
1/2<r≤1C±r , K f−2
c (Ωc \ fc(Ωc)) =⋃k1∈Z(E+
k1∪ E−k1), Ù¥ E±k1 =
⋃1/2<r≤1C
±r,k1
,
fc(C+r,k1
(t) ∪ C−r,k1(t)) = C+r (t) ∪ C−r (t) t ∈ R, E±k1 >.¹ C±1 (t0 + 2k1π). 5¿
f−1c (Ωc)´ëÏ,K C\f−1
c (Ωc)©|Ñ´üëÏ. AO/, C\f−1c (Ωc) = Y +∪Y −,
Ù¥ Y +Ú Y −L« C\f−1c (Ωc)©Oá3 f−1
c (Ωc)þ¡Úe¡©|. N´wÑ f−2c (Ωc)
Ó´üëÏ, Ïd C \ f−2c (Ωc) =
⋃k1∈Z(Y +
k1∪ Y −k1 ), Ù¥ Y +
k1(½ Y −k1 ) E+
k1(½
E−k1)>.. 5¿´ E+ Ú Y + (½ E− Ú Y −) u f−1c (fc(Ωc))þ(½e).
, , E+k1Ú Y +
k1(½ E−k1 Ú Y −k1 ) ±Ø´ù.
éu 1 ≤ i ≤ 4, 1/2 < r ≤ 1, n ≥ 2 Ú k1, · · · , kn ∈ Z, ·±8B/½
 D±i,k1,··· ,kn , C±r,k1,··· ,kn , E±k1,··· ,kn ±9 Y ±k1,··· ,kn . ¯¢þ, b D±i,k1,··· ,kn−1, C±r,k1,··· ,kn−1
,
E±k1,··· ,kn−1Ú Y ±k1,··· ,kn−1
®²½ÂÐ. Ï fc 3 Y ±k1,··· ,kn−1þ´ü, 3
D±i,k1,··· ,kn , C±r,k1,··· ,kn , E±k1,··· ,kn Ú Y ±k1,··· ,kn ¦
(1) D±i,k1,··· ,kn , C±r,k1,··· ,kn , E±k1,··· ,kn , Y ±k1,··· ,kn ѹ3 Y ±k1,··· ,kn−1¥;
(2) fc(⋃k1∈ZD
+i,k1,··· ,kn ∪D
−i,k1,··· ,kn) = D+
i,k2,··· ,kn ∪D−i,k2,··· ,kn ;
(3) fc(⋃k1∈ZC
+r,k1,··· ,kn ∪ C
−r,k1,··· ,kn) = C+
r,k2,··· ,kn ∪ C−r,k2,··· ,kn ;
(4) fc(⋃k1∈ZE
+k1,··· ,kn ∪ E
−k1,··· ,kn) = E+
k2,··· ,kn ∪ E−k2,··· ,kn ;
(5) fc(⋃k1∈Z Y
+k1,··· ,kn ∪ Y
−k1,··· ,kn) = Y +
k2,··· ,kn ∪ Y−k2,··· ,kn .
éu?¿ n ≥ 2, ·k f−nc (Ωc) \ f−n+1c (Ωc) =
⋃1≤i≤4, k1,··· ,kn−1∈ZD
±i,k1,··· ,kn−1
∪E±k1,··· ,kn−1
C \ f−nc (Ωc) =⋃k1,··· ,kn−1∈Z Y
±k1,··· ,kn−1
.
éz c ∈ Σ1/2, 5¿ φc : Ωc → D ¦ φ′c(0) > 0 5zN φc Û
¼ê. éu k 6= 0, P
γk := c ∈ Σ1/2 : ∂Ωc ∩ Critc = π/2,−π/2, c+ kπ,−c− kπ \ ±π/2 (5.3.3)
γ0 := γ. é?¿ c ∈ γk0 (XJù c 3), Ï φc : Ωc → D ±Ó/ò
§5.3. 5z« 83
ÿ φc : Ωc → D, ·½Â π/2 Ú c+ k0π mÝ
A(c) = argφc(c+ k0π)/φc(π/2), (5.3.4)
Ù¥ 0 ≤ arg(z) < 2π ˼ê. AO/, XJ c ∈ γ K A(c) = argφc(c)/φc(π/2), A(π/2) = 0, A(−π/2) = π. d, c 3 γ þCz, A(c) ëY6u c. XJ3,
k0 6= 0 ¦ γk0 6= ∅, K3 c ∈ γk0 ¦ A(c) = argφc(c+ k0π)/φc(π/2) 6= 0 Ú π.
âcÓ©Û, c ∈ γ , fc fc kÓ|Ü&E. ½öoÑ/`, fc Äå
XÚaqu fc 3ã 5.1 ¥¤«¹.
·K 5.3.11. XJ A(c1) = A(c2), Ù¥ c1, c2 ∈⋃k∈Z γk, K c1 = c2.
y². XJ c1 Ú c2 ¥ku π/2 ½ −π/2, K?Ø´². e5, ·
b§üÑØu π/2 Ú −π/2. Ϥk.:Ñ:¤áÚ, Kéz
c ∈⋃k∈Z γk, fc Fatou 8=3;.
Ø5, b c1 Ú c2 Ñáu γ. P φ = φ−1c2 R% φc1 : Ωc1 → Ωc2 , Ù¥
R%(z) = e2πi% L«f5^= % = arg(φc2(π/2)/φc1(π/2))/2π. Ï A(c1) = A(c2), /
N φ3 Ωc1 þò fc1 Ý fc2 ,÷v φ(π/2) = π/2Ú φ(c1) = c2. - ϕ0 : C→ Cφ ¦ ϕ0(∞) = ∞ [/*Ü. ùo´UÏ>. ∂Ωc ´[ (
[43, ½n 8.2]). ·ò^ fc ÄåXÚ8B/½Â[/N ϕn : C→ C.
üå, ·òÎÒ fcj Ú Ωcj ¥eI cj ^ j O, Ù¥ j = 1, 2. âcé
fc |Ü&E©Û, ùL²éz©| Dkj (c1) Ú Ωk
1, 3éA Dkj (c2) Ú Ωk
2,
Ù¥ j = 1, 2 k ∈ Z. 3 Ω1 þ- ϕ1 = ϕ0 = φ, ÏLÀJ f2 ·_, ·±½
 ϕ1 = f−12 ϕ0 f1, §ò D
k
j (c1) Ú Ωk
1 ©ON Dk
j (c2) Ú Ωk
2.
éu?Û z ∈ f−11 (Ω1 \ f1(Ω1)) = E+(c1) ∪ E−(c1), ½Â ϕ1 = f−1
2 ϕ0 f1, À
J_©| f−12 ¦XJ z = C+
r (t)(c1) ½ C−r (t)(c1), Ù¥ t ∈ R 1/2 < r ≤ 1,
K ϕ1(z) = C+r (t)(c2) ½ C−r (t)(c2). u ϕ1 3>. f−1
1 (f(∂Ω1)) þëY5, ùL²
ϕ1 : f−11 (Ω1)→ f−1
2 (Ω2) ´X.
·Iò ϕ1 òÿ Cþ[/N. Ï C\f−11 (Ω1)ü©| Y +(c1)
Ú Y −(c1) Ñ´üëÏ3 Ω2 ¡Ø3 f2 ., 3 C \ f−11 (Ω1) þ·½Â
ϕ1 ϕ1 = f−12 ϕ0 f1 : Y ±(c1)→ Y ±(c2). N´wÑ ϕ1 : C→ C ´[/N
ϕ1 Ú ϕ0 û÷v ‖µϕ1‖ = ‖µϕ0‖.béz 1 ≤ k ≤ n−1,3[/N ϕk : C→ C¦ f2 ϕk = ϕk−1 f1
ϕk : f−k1 (Ω1)→ f−k2 (Ω2) ´X. y3·Xe8B/½Â ϕn.
3 f−n+11 (Ω1) þ½Â ϕn = ϕn−1. éz D±i,k1,··· ,kn−1
(c1) ⊂ f−n1 (f1(Ω1)) \ f−n+11 (Ω1),
3éAD±i,k1,··· ,kn−1(c2) ⊂ f−n2 (f2(Ω2))\f−n+1
2 (Ω2),Ù¥ 1 ≤ i ≤ 4 k1, · · · , kn−1
∈ Z. ÏLÀJ f2·_,½Â ϕn = f−12 ϕn−1f1 : D
±i,k1,··· ,kn−1
(c1)→ D±i,k1,··· ,kn−1
(c2).
84 1ÊÙ x¼êÄåXÚ
é?Û z ∈ f−n1 (Ω1 \ f1(Ω1)) = E+k1,··· ,kn−1
(c1) ∪ E−k1,··· ,kn−1(c1), ½Â ϕ1 = f−1
2 ϕn−1 f1. ÀJ_ f−1
2 ¦XJ z = C+r,k1,··· ,kn−1
(t)(c1)½ö C−r,k1,··· ,kn−1(t)(c1),Ù¥ t ∈ R
1/2 < r ≤ 1, K ϕn(z) = C+r,k1,··· ,kn−1
(t)(c2) ½ C−r,k1,··· ,kn−1(t)(c2). u ϕ1 3>.
f−n1 (f1(∂Ω1)) þëY5, ùL²·kl f−n1 (Ω1) f−n2 (Ω2) )ÛN ϕn.
5¿ C \ f−nc (Ωc) =⋃k1,··· ,kn−1∈Z Y
+k1,··· ,kn−1
∪ Y −k1,··· ,kn−1. - Y +
k1,··· ,kn−1(c1) C \
f−n1 (Ω1) ©| (Y −k1,··· ,kn−1aq?Ø), 3éA©| Y +
k1,··· ,kn−1(c2) ⊂ C \
f−n2 (Ω2). Ï Y +k1,··· ,kn−1
´üëÏع., 3J,N ϕn : Y +k1,··· ,kn−1
(c1) →Y +k1,··· ,kn−1
(c2) ¦ f2 ϕn = ϕn−1 f1.
3ù«¹e, ·[/N ϕn : C → C ¦ f2 ϕn = ϕn−1 f1
‖µϕn‖ = ‖µϕn−1‖. â8BL§, ·l C NgC[/N ϕnn≥0 ¦
ϕn 3 f−n1 (Ω1) þ´/ ϕn Beltrami Xê÷véz n ≥ 1 k ‖µϕn‖∞ =
‖µϕn−1‖∞. ùL²3~ê 0 < k < 1, ¦ ‖µϕn‖∞ ≤ k < 1 é¤k n ≥ 0 ¤á.
ÀJ ϕnn≥0 Âñfüg, ·ü4[/N ϕ Ú ψ. §± 0,
π/2Ú∞ØÄ,÷v f2ϕ(z) = ψf1(z),ù´Ïéz n ≥ 1k f2ϕn = ϕn−1f1.
âE, 3⋃n≥0 f
−n1 (Ω1) þk ϕ(z) = ψ(z). Ï fc Fatou 83 C þ´È
, ϕ = ψ 3E²¡Èf8þ¤á. âëY5, ùL² ϕ ψ 3 C þ. u´ f1 Ú f2 3 C þ´[/Ý3 Fatou 8þ´/Ý.
â [45, ·K 1.14], éu Julia 8¥A¤k:, Ù ω-48o¹3ÛÉ
c;¥, o<ºÃ¡. y3ÏÛÉc;á3 Fatou 8¥, ùL²
Julia 83"ÿÝ¿Âeu<º8. â [64] ¥½n 1.2, z f ∈ Σ1/2 3
<º8þvkØC. ùL²z f ∈ Σ1/2 3 Julia 8þvkØC.
·^aqu [56]¥?Ø5y² f1 3E²¡þ/Ýu f2. - µϕ
[/N ϕEA,Ï ϕò f1Ý f2,Ïd µϕ´ f1–ØC. =, f ∗1 (µϕ) = µϕ.
OL² |µϕ(f1(z))| = |µϕ(z)|. b |µϕ(z)| > 0 3 E þ¤á, ù E ´ f1
äkÿÝf8, u´·k |µϕ(z)/|µϕ(z)‖ = 1, µϕ(z)/|µϕ(z)| 3 E þ
´ f1–ØC. ùL² µϕ(z)
|µϕ(z)|dzdz´ J(f1) þØC, ù´gñ. u´ µϕ = 0
3 C þA??¤á, ϕ 3 C ´/. Ï ϕ ± 0, π/2 Ú ∞ ØÄ, ù`² ϕ
´ðNl f1 = f2.
íØ 5.3.12. γ = c ∈ Σ1/2 : ∂Ωc ∩ Critc = π/2,−π/2, c,−c. é?¿ k 6= 0,
k γk = ∅.
y². âÚn 5.3.10, γ´^²L π/2Ú −π/2 Jordan.Ï A(π/2) = 0,
A(−π/2) = π A(c) ëY6u c, ùL² A : γ → [0, 2π) ´÷. XJ3 c 6∈ γ ¦ ∂Ωc TвL4 .: ±π/2 Ú ±c, K§·K 5.3.11 gñ. 1(رd1
§5.4. äk û[ 85
(ØÚ·K 5.3.11 íÑ.
·K 5.3.11 ÚíØ 5.3.12 w·ëê c ∈ γ dÝ A(c) û½, ùÝ¡
´ π/2 Ú c m “/Ý”.
½n 5.3.5 Ú 5.1.2 y². XJ3 c ∈ Σ1/2 ¦ ∂Ωc TвLü.:
±(c + k0π), Ù¥ k0 6= 0. ÀJ^ëY η : (0, 1) → Σ1/2 ¦ limt→0 η(t) = c
limt→1 η(t) =∞. âÚn 5.3.9, cá3∞S, ∂ΩcTвLü.
: π/2Ú −π/2. âëY5,3 c ∈ η ¦ ∂Ωc ²L 4.: ±π/2Ú ±(c+k0π),
ùíØ 5.3.12 1(Øgñ. Ï XJ c ∈ γext, ∂Ωc UTвLü.:
π/2 Ú −π/2. u´(½n±d γ ½Â, Ún 5.3.10 ÚíØ 5.3.12 íÑ.
§5.4 äk û[
®² ∂Ωc ´[, y²ÃâN´ëY, ·I`²>. ∂Ωc
äk û.
5¿ c 3 Σ1/2 ¥ªu ∞ , = Im(c) → ∞ , ¼ê fc ÛÜÂñu
f∞(z) = sin(z)/2. Ïdéu ∞ 3 Σ1/2 ?Û4, ·±ò ∞ \ù4þ, ¦ ∞ #C¤ Σ1/2 ∪ f∞ ;f8.
XJ c á3 0 S, 5¿ c ªu 0 , L−1c fc Lc(z) ÛÜ
/Âñngõª f0(z) = z/2− z3/6, Ù¥ Lc(z) = cz. |^Óg, ·
0 3 L−1c fc Lc(z) : c ∈ Σ1/2 ∪ f0 ¥;. Ï Ωc ëY6u c, |^k
CX, ·k
Ún 5.4.1. 3~ê M > 0 ¦éu c ∈ Σ1/2 k Ωc ⊂ DM .
e5·y²
½n 5.4.2. >. ∂Ωc ´ K–[, ~ê K Ø6u c ∈ Σ1/2.
y². y²g´Äu [41]. ·=Ié c ∈ Σ1/2\±π/2y² ∂Ωc ´ K–
[. éü¢ê a = a(c)Ú b = b(c),·^ a b5L« a ≤ Cb,Ù¥ C > 0´=
6u c~ê. é?¿½ c ∈ Σ1/2ÚØÓ x, y ∈ ∂Ωc,- L = [x, y]ë xÚ
y ã, I, J ∂Ωc\x, yü¦ |fc(I)| ≤ |fc(J)|ëÏ©|. âÚn 2.4.1
(- r = 1/2) Ú»½Â, ·k |fc(I)| |fc(x)− fc(y)| ≤ Diam(fc(L)) ≤ |fc(L)|.éu½ r > 1 Ú Diam(I)/(8r) ≤ m ≤ (8r − 1)Diam(I)/(8r), ½Â
Ar = z : m ≤ |z − x| ≤ m+ Diam(I)/(8r).
86 1ÊÙ x¼êÄåXÚ
- I1 Ar ∩ I ¦ |I1| ≥ Diam(I)/(8r) ©|. Ï I1 Ú L ;8, 3
u ∈ I1 Ú v ∈ L ¦ |f ′c(u)| = minz∈I1 |f ′c(z)| |f ′c(v)| = maxz∈L |f ′c(z)|. Ïd·k |fc(L)| ≤ |f ′c(v)||L| Ïd Diam(I)|f ′c(u)|/(8r) ≤ |f ′c(u)||I1| ≤ |fc(I1)| ≤ |fc(I)| |fc(L)| ≤ |f ′c(v)||L|, ùL² Diam(I)/|L| |f ′c(v)|/|f ′c(u)|. y²ù½n, ·I
|f ′c(v)|/|f ′c(u)| þ., ½ö*/w, u ATl.:.
5¿ Critc fc.:8Ü.·äó3· rÚm¦é?¿ w ∈ Critc,
k |u − w| ≥ Diam(I)/(8r). ¯¢þ, âÚn 5.4.1, D(x,Diam(I)) ⊂ DM
õ¹kõ, b´ N > 0 .:. - r = N , é k = 1, · · · , 4r, ÄAk = z : (2k − 1) Diam(I)/(8r) ≤ |z − x| ≤ 2kDiam(I)/(8r) . 3 Ak0 ¦
é?¿ u ∈ Ak0 , k |u− w| ≥ Diam(I)/(8r) éz w ∈ Critc ¤á.
y3Ä Ar, Ù¥ r = N m = (2k0 − 1) Diam(I)/(8r). Ï u ∈ I1 ⊂ Ar
v ∈ L, K |u − x| ≤ Diam(I) |v − x| ≤ |L| ≤ Diam(I). Ïd |u − v| ≤ 2 Diam(I) ≤16r |u − w| éz w ∈ Critc ¤á. l |v − w| ≤ |v − u| + |u − w| |u − w|, ùL²|v − w|/|u− w| 1.
3~ê a > 0 ¦é?¿ w ∈ Critc, k | sin(z − w)|/|z − w| ≤ a é
z ∈ DM ¤á, Ù¥ DM ´Ún 5.4.1 ¥½Â. éu k ∈ Z, P Hk = z : kπ ≤Re(z) < (k + 1)π. XJ Re(c) ∈ [π/4, π/2], c 6= π/2 u ∈ Hku , Ù¥ ku ∈ Z 6uu. K Crit′c := Hku ∩ Critc = c + kuπ, π/2 + kuπ,−c + (ku + 1)π. 5¿ Crit′c ⊂ z :
π/4 + kuπ ≤ Re(z) ≤ 3π/4 + kuπ Critc \ Crit′c Hku ålu π/4, ùL²3
~ê b > 0 ¦ | sin(z − w′)|/|z − w′| ≥ b é w′ ∈ Crit′c Ú z ∈ Hku ∩ DM ¤á. Ï
|v − w|/|u− w| 1 é w ∈ Critc ¤á, ·k
| sin(v − w′)|| sin(u− w′)|
=|v − w′||u− w′|
| sin(v − w′)|/|v − w′|| sin(u− w′)|/|u− w′|
≤ |v − w′|
|u− w′|a
b 1. (5.4.1)
5¿ f ′c(z) = sin(z−π/2) sin(z+c) sin(z−c)/(2 sin2(c)) = sin(z−w′1) sin(z−w′2) sin(z−w′3)/(2 sin2(c)), Ù¥ Crit′c = w′1, w′2, w′3, l â (5.4.1) ·k |f ′c(v)|/|f ′c(u)| 1.
ùL² Diam(I)/|L| 1.
XJ Re(c) ∈ [0, π/4), ½Â Hku = z : −3π/8 + kuπ ≤ Re(z) < 5π/8 + kuπ. |^þãaq?Ø, ·Ók Diam(I)/|L| 1. XJ Re(c) ∈ [−π/2, 0), K
Re(−c) ∈ (0, π/2]. 5¿ fc = f−c. ½ny..
§5.5 ÃâNEÚ§ëY5
ù!¥, ·½Âl Σ1/2 Σλ ÃâN S ¿y²§ëY5. Ìg
´ÏLE.Nò5z« Ωc =¤ Siegel . >. ∂Ωc [
§5.5. ÃâNEÚ§ëY5 87
/Ýu Siegel >.. éuEù.N©, [41, 79].
- γint Ú γext ©OL« Σ1/2 \ γ k.ÚÃ.©| (5¿ γext dü©||¤).
â½n 5.3.5 ¥¤ã Σ1/2 (½n, e5, ·=r5¿å8¥3Ã.©| γext
þ, éu γint ±?1aq?Ø.
éz c ∈ γext = γext ∪ γ ∪ f∞, Ù¥ f∞(z) = sin(z)/2, · fc(∂Ωc) ´
¢)Û ∂Ωc ¹üé¡.: ±π/2. P Uc = C \ Ωc, Vc = C \ fc(Ωc)
vc = fc(π/2) ∈ fc(∂Ωc). âÿ Riemann N½n, éz ξ ∈ ∂Ωc, 3
/N hξc : Vc → Uc ¦ hξc(∞) =∞ hξc(vc) = ξ. du Uc Ú Vc 'u:é¡,
âÚn 5.3.3 ¥aq?Ø, hξc ´Û¼ê hξc(−vc) = −ξ. ξ 3 ∂Ωc þ$Ä, N hξc fc|∂Ωc ´l ∂Ωc gCëYÚüN±Ó
x. â [39] ¥·K 11.1.9, 3 ξc ∈ ∂Ωc, ¦hξcc fc|∂Ωc ^=ê´·3
c½Ãnê θ. üå, ·P hξcc hc.
- ϕc : C \D→ Uc ÷v ϕc(∞) =∞ ϕc(1) = π/2 Riemann N. aquk
c?Ø, ϕc ´Û¼ê. P sc = ϕ−1c hc fc ϕc, K sc|S1 : S1 → S1 ´^=ê θ
.±Ó. e¡Úny²~aqu [41] Ún 4.2 Ú [79] ¥Ún 8.
Ún 5.5.1. 3¹ S1 m A, ¦éu?¿ c ∈ γext, sc :
S1 → S1 ±)Ûòÿ A.
y². 5¿ γext ´;. Iy²éuz c0 ∈ γext, 3 c0 3 γext ¥
m D(c0) ±9 S1 m A(c0), ¦éz c ∈ D(c0), sc ±)Ûòÿ
A(c0) =.
Äk·b c ∈ γext ∪ f∞, Ïd ∂Ωc Tйü.: ±π/2. ?Øò¬©¤
ü«¹.
1«¹, é?Û z ∈ S1 Ú z 6= 1,−1, · sc 3 z l¡N3 Dþ N ′ = D(z, ε) \ D S´X. ùp, ε v¦ sc ò N ′ Ó/N
sc(N′), §Ó´ sc(z) l¡N3 D þ. âÚn 5.2.1, sc ±
X/òÿ z m N þ, ¿¦ sc ò N Ó/N÷ sc(z) ∈ S1
m sc(N) þ.
1«¹,XJ z = 1 (aq?ر^u z = −1),- sc = (ϕ−1c hc) (fc ϕc).
·Äkl D¡ÀJ 1W ′,, fcϕc òW ′ N÷ÿÀ
fcϕc(W ′),ò)ÛãW ′∩S1 Ó/N ∂Vc þ^)Ûã. ùp, fcϕc|W ′Ø´_;. ¯¢þ, W ′ ±©¤nÜ©,Ù¥üÜ© fc ϕc N fc ϕc(W ′)∩Vc, eÜ©N fc ϕc(W ′) \ Vc. ÓâÚn 5.2.1, fc ϕc|W ′ ±òÿ W ′
m, § 3 : 1 /N fc ϕc(1) m. Ï ∂Vc ´)Û, 3
88 1ÊÙ x¼êÄåXÚ
fc ϕc(1) l¡N3 ∂Vc þ N ′1, ¦ ϕ−1c hc X/òÿ N ′1
m N1, ò N1 Ó/N ϕ−1c hc(N1). ¿© N ′1, ¦ N ′1 ⊂ fc ϕc(W ′).
ùL² sc ±)Û/òÿ± 1 %mþ.
Ï fc, hc Ú ϕc ëY6u c, · sc ëY6u c. Ïdéz c0 ∈γext ∪ f∞ Ú z ∈ S1, 3 γext ¥m D(c0), ±9 z m B(z)
¦éz c ∈ D(c0), sc|B(z)∩S1 ±X/òÿ B(z)þ. Ï S1 ´;, 3 S1
m A(c0) ¦éz c ∈ D(c0), sc|S1 ±X/òÿ A(c0) þ.
XJ c ∈ γ, þ¡òÿL§±aq/?1, ´dIÄüé.: ±1 Ú
±ϕc(c). Ï γext ´;, 3kõ D(ci), i = 1, · · · , n, ¦ γext §¿CX.
- A = A(c1) ∩ · · · ∩ A(cn), Ké?Û c ∈ γext, sc ±)Û/òÿ A þ.
5 5.5.2. N sc ØUòÿ ±1 S.
dÚn 5.5.1, · sc ´äkk..^=ê θ ¢)Û.±Ó. â
½n 5.2.2, ·k
Ún 5.5.3. 3 k1 > 0,¦éz c ∈ γext,k[é¡Ó pc : S1 → S1
÷v p−1c sc pc = Rθ, pc(1) = 1 K(pc) ≤ k1, Ù¥ K(pc) pc [é¡~ê.
Ún 5.5.4. 3~ê K1 ≥ 1, ¦éz c ∈ γext, hc : Vc → Uc, ϕc : C \D→Uc Ú pc : S1 → S1 ©Ok[/*Ü Hc : C → C, Φc : C → C Ú Pc : D → D,
Hc, Φc Ú Pc ûÑØL~ê K1. d,§ÑÛ¼ê. AO/, Hc(0) = 0,
Φc(0) = 0 Pc(0) = 0.
y². 5¿ φc : Ωc → D \5^ φ′c(0) > 0 5z¼ê, §ëY6u c.
P rc = φc hc φ−1c L1/2 : S1 → S1, K rc(−z) = −rc(z), ù´Ï hc Ú φc Ñ´Û¼ê,
Ù¥ La(z) = az ´5N. d, rc ´[é¡Ó, ù´Ï ∂Ωc Ú f(∂Ωc)
Ñ´[.
N rc 3 τ(x) = e2πix eJ, Rc : R → R ÷v Rc(x + 1/2) = Rc(x) + 1/2.
âúª (5.2.2), ùL² Beurling-Ahlfors *Ü (Rc)BA : H → H Ó÷v (Rc)BA(z +
1/2) = (Rc)BA(z) + 1/2. u´ Beurling-Ahlfors *Ü (rc)BA : D → D ÷v (rc)BA(−z) =
−(rc)BA(z).
y3é z ∈ Vc ½Â Hc(z) = hc(z) é z ∈ fc(Ωc) ½Â Hc(z) = φ−1c (rc)BA L2
φc(z). K Hc ´ hc [/*Ü. w,, Hc ´Û¼ê Hc(0) = 0.
â½n 5.4.2, éu c ∈ γext, ∂Ωc K–[, Ù¥ K ´Ø6u c ~ê.
ùL²éu¤k c ∈ γext, N rc [é¡~ê=6u K. ù¿X Hc û
Ø6u c ∈ γext.
§5.5. ÃâNEÚ§ëY5 89
aq?ر^uòÿ ϕc Ú pc (|^Ún 5.5.3). y²..
5 5.5.5. Ún 5.5.4 ¥^òÿ¡5z (Beurling-Ahlfors) [/ò
ÿ. k, Douady-Earle òÿ [27] ´ÐÀJ. N´wÑ Hc, Φc Ú Pc ëY6
u c.
½Â
Fc(z) =
Φc Pc Rθ P−1
c Φ−1c (z − kπ) éu z ∈ Ωk
c , k óê,
−Φc Pc Rθ P−1c Φ−1
c (z − kπ) éu z ∈ Ωkc , k Ûê,
Hc fc(z) Ù§¹.
â Fc ½ÂÚÚn 5.5.4, y
Ún 5.5.6. Fc ´ëY, ±Ï 2π ±ÏÛ¼ê÷v Fc(z + π) = −Fc(z). d
, Fc ": kπ,± arcsin(√
3 sin(c)) + kπ : k ∈ Z.
y3·±3 C þ½ÂXe Fc–ØC Beltrami Xê νc(z) :
νc(z) =
(P−1
c Φ−1c )∗σ0 éu z ∈ Ωc,
(F nc )∗(P−1c Φ−1
c )∗σ0 éu z ∈ F−nc (Ωc) \ F−n+1c (Ωc), n ≥ 1,
σ0 Ù§¹.
Ï Fc3 C\⋃k∈Z Ωk
c þ´)Û,âÚn 5.5.4,ùL²3½~ê k ∈ (0, 1),
¦éz c ∈ γext, Beltrami Xê νc ÷v ‖νc‖∞ < k. âÿ Riemann N½n,
3± 0, 2π Ú ∞ ØÄ[/N wc : C → C, ¦§´k Beltrami Xê
νc Beltrami §). d, wc EAØLØ6u c ~ê K2.
Ún 5.5.7. [/N wc ´Û¼ê, wc(z + π) = wc(z) + π wc(π/2) = π/2.
y². Ï Fc ´Û¼ê Fc(z + π) = −Fc(z), â νc ½Â, k νc(z) = νc(−z)
νc(z + π) = νc(z). =, wc(z + π) Ú wc(z) äkÓ Beltrami Xê. ϧÑ
± ∞ ØÄ, K3, a Ú b ¦ wc(z + π) = awc(z) + b.
XJ |a| < 1, Ï wc ± 0 ØÄ, ·k wc(kπ) = awc(kπ − π) + b = · · · =
b(1 + a + · · · + ak−1) = b(1 − ak)/(1 − a) éz k ≥ 1 ¤á. u´ k ªu +∞ wc(kπ) ªu b/(1− a). aq/, XJ |a| > 1, Ï wc(z) = (wc(z + π)− b)/a, ·k
wc(−kπ) = b(1− 1/ak)/(1− a), Ù¥ k ≥ 1. ùL² k ªu +∞ wc(−kπ) Óª
u b/(1− a). ùü«¹Ñ´ØUÏ wc ± ∞ ØÄ.
XJ a = e2πix, Ù¥ x = m/n ∈ (0, 1) ´knê m,n *dp, Kéz
k ≥ 0 k wc(kn) = 0. ù´ØU. XJ a = e2πix, Ù¥ 0 < x < 1 ´Ãnê,
90 1ÊÙ x¼êÄåXÚ
âÚn 2.4.3, 3 N f knn≥1, ¦ n ªu ∞ e2πknxi ªu 1. ù
Ó wc(∞) =∞ gñ. Ïd~ê a 7Lu 1. Ï wc(2π) = (1 + a)b = 2π, ùL²
b = π.
P wc(z) = −wc(−z), Ï wc(0) = wc(0) wc Beltrami Xê νc , ùL
²3, d ¦ wc(z) = dwc(z). u´ wc(π) = −wc(−π) = π = dwc(π) = dπ. ùL²
d = 1 wc ´Û¼ê. (Øâ wc(π/2) = wc(−π/2) + π = −wc(π/2) + π
.
Ï νc ´ Fc–ØC, · Tc(z) = wc Fc w−1c (z) ´ C gC)ÛÛ¼
ê, T¼ê3:äk^=ê θ Siegel . âE, ù Siegel >.´
²L Tc é.:[.
`² Tc ∈ Σλ, ·IÄ Tc ":¿|^Ún 5.2.5. âÚn 5.5.6, ·
k
Ún 5.5.8. Tc = wc Fc w−1c ´Û¼ê Tc(z + π) = −Tc(z). d, Tc ":8
´ kπ,±wc(arcsin(√
3 sin(c))) + kπ : k ∈ Z.
íØ 5.5.9. Tc ∈ Σλ.
y². 5¿ Tc k u:^=ê θ Siegel. âÚn 5.2.5Ú 5.5.8
(J, ·I`² Tc ´k.
ĽÂ3:NC[/N 1/wc(1/z). y¿â Mori ½n [43, ½
n 3.2], O |wc(z)| ≤ 16K2 |z|K2 Ú |w−1c (z)| ≤ 16K2|z|K2 , Ù¥ z 3 ∞ SC
z K2 ´ cÃ'~ê, ù´Ïé?¿ c ∈ γext, wc ´ K2–[/N.
âÚn 5.5.4, 3 ∞ S, k |Hc(z)| ≤ 16K1 |z|K1 . â Fc 3 ∞ S½Â, k |Tc(z)| ≤ C(K1, K2)e|z|
K2 , Ù¥ C(K1, K2) ´=6u K1 Ú K2 ~ê.
ùL² Tc ´k.
y3·½ÂÃâN S : γext 7→ Σλ S(fc) = Tc. daq?Ø, S Ó±½Â3 γint þ. AO/, S(f∞) = T∞ S(f0) = T0, Ù¥ T∞(z) = e2πiθ sin(z)
T0(z) = e2πiθ(z − z3/3).
5 5.5.10. /, Ø c = π/2 ½ −π/2, k c 6= wc(c), Ïd·¿ØU Tc =
2e2πiθfc. c = π/2 ½ −π/2 , ÙéA Σλ ¥¼ê z 7→ e2πiθ(sin(z)− sin3(z)/3)
u: Siegel >.´²Lüê 3 .: π/2 Ú −π/2 [.
ù(Jaqu [79] ¥Ì½n.
§5.5. ÃâNEÚ§ëY5 91
y²ÃâN´ëY, âÚn 5.2.5 Ú 5.5.8 ±9íØ 5.5.9, ·Iy²
":N c 7→ wc(arcsin(√
3 sin(c))) ëY6u c. Äk·`²
Ún 5.5.11. N c 7→ νc ´ëY.
y². - cn Âñ c∞ ∈ γext S, ·I`² n→∞ , νcn 3 L∞
¥ÓÂñ νc∞ . e5, üå, ·^ n Ú ∞ ©OOeI cn Ú c∞.
E C¥ Lebesgueÿ8,- Area(E)L« E 3¥¡Ýþe Lebesgue
¡È. éu δ > 0, ½Â
Q δn = z ∈ C : |νn(z)− ν∞(z)| > δ.
- Ekn =
⋃0≤i<k F
−in (Ωn), Ù¥ 1 ≤ n, k ≤ ∞. ·^ En Ú E∞ 5L« E∞n Ú E∞∞ , §
©OéAX Ωn Ú Ω∞ ;. â νc ½Â, k Q δn ⊂ En ∪E∞, ù´Ï3Ù§
/ νn = ν∞ = 0. Iy²éu?¿ δ > 0 Ú ε > 0, 3¿© N ¦éz
n > N , k Area(Q δn) < ε.
½ δ > 0 Ú ε > 0, Ï Ωn ´ Fn–ØC, XJ k v, K Area(En \Ekn) ±
v. Ï Ωc ëY6u c, 3¿© k0 ¦é¤k n,
Area(En \ Ek0n ) < ε/5 and Area(E∞ \ Ek0
∞) < ε/5. (5.5.1)
âëY5,3mÿÀ Ω′ Ú N1 > 0,¦é¤k n > N1, Ω′ ⊂ Ωn∩Ω∞
Area(Ek0n \Dk0
∞) < ε/5 and Area(Ek0∞ \Dk0
∞) < ε/5, (5.5.2)
Ù¥ Dk0∞ =
⋃0≤i<k0 F
−i∞ (Ω′).
âÚn 5.5.4 Ú5 5.5.5, n ªu ∞ , [/N (Φn Pn)−1 3 Ω′ þ
Âñu (Φ∞ P∞)−1. ,, n ªu ∞ , (Φn Pn)−1 F k0n 3 Dk0∞ þÓÂ
ñu (Φ∞ P∞)−1 F k0∞ . ùL² νn 3 Dk0∞ þA??Âñu ν∞. Ïd, 3 N2 > 0
¦é¤k n > N2, ·k
Area(Q δn ∩Dk0
∞) < ε/5. (5.5.3)
- N = maxN1, N2, â (5.5.1), (5.5.2) Ú (5.5.3), ùL²é¤k n > N , k
Area(Q δn) < ε. ùy² c 7→ νc 3 c∞ ?´ëY. â c∞ ?¿5, Úny.
íØ 5.5.12. ÃâN S : γext → Σλ ´ëY.
y². âÚn 5.5.11, · νc ëY6u c. Ï wc ± 0, 2π Ú ∞ ØÄ ‖νc‖∞ ≤ k < 1 éz c ∈ γext ¤á, â Ahlfors Ú Bers ½n, ù`² wc ëY
6u c . âÚn 5.5.8, Tc ":ëY6u c. âÚn 5.2.5, ùL² S ëY6uc ∈ γext.
92 1ÊÙ x¼êÄåXÚ
§5.6 ½n 5.1.1 y²
5¿ γ ëêm Σ1/2 ¥½Â¦éuz c ∈ γ \ ±π/2, 5z«>. ∂Ωc Tй fc 4 .:8Ü. éz c ∈ γ, - ∆S(c) S(c) ∈ Σλ ±
:% Siegel , Ù¥ S ÃâN. Ø c = ±π/2 , >. ∂∆S(c) ´T
вLüé.: ±π/2 Ú ±S(c) [.
éu c ∈ Λ, ·^ gc 5P Σλ ¥¼ê. - ϕS(c) : ∆S(c) → D ¦ ϕS(c) gS(c) ϕ−1S(c) = Rθ Ú ϕS(c)(π/2) = 1 5zN. aquÚn 5.3.3 ?Ø, ϕS(c) ´Û¼
ê. ½Â π/2 Ú S(c) mÝ
B(S(c)) = argϕS(c)(S(c)), (5.6.1)
Ù¥ 0 ≤ arg(z) < 2π ˼ê. AO/, B(S(π/2)) = B(π/2) = 0 B(S(−π/2)) =
B(−π/2) = π. Ï S ëY6u c, ùL² B(S(c)) ÓXd.
Ún 5.6.1. XJ B(S(c1)) = B(S(c2)), Ù¥ c1, c2 ∈ γ, K S(c1) = S(c2).
ÏÚn 5.6.1y²~aqu·K 5.3.11y²,·=Ñy²V.Äk,
·If5Ún, §aqu [77, íØ 5.2] Ú [41, Ún 5.1].
Ún 5.6.2. z gc 3 Σλ ¥[/Ýa´m8½ö´ü:. XJ
c ∈ γ, K gS(c) [/Ýa´ü:.
y². b c, c′ ∈ Σλ c 6= c′, [/N ϕ : C → C ÷v ϕ gc ϕ−1 = gc′ .
- µϕ L« ϕ Beltrami Xê, §´ gc–ØC. =, µϕ(gc(z))g′c(z)/g′c(z) = µϕ(z), d
/, tµϕ(gc(z))g′c(z)/g′c(z) = tµϕ(z), ùL² Beltrami Xê tµϕ Ó´ gc–ØC, Ù¥
|t| < 1/‖µϕ‖∞. Päk Beltrami Xê tµϕ Beltrami §) ϕt, K ϕt gc ϕ−1t Ñ
´)Û.
aquíØ 5.5.9 ?Ø, z ϕt gc ϕ−1t Ñáu Σλ. P ϕt gc ϕ−1
t
gc(t), K§X6u t, ù´Ï Beltrami Xê tµϕ X6u t. u´ t 7→ c(t) 3
|t| < 1/‖µϕ‖∞ S´X. Ïd[/Ýao´m8oü:.
XJ c ∈ γ \ ±π/2, gS(c) Siegel >.Tй 4 .:. 3 c ?Û
S, Ñ3 c′ ¦ gS(c′) Siegel >.=¹üé¡.:. ùL² gS(c) Ú
gS(c′) $ÑØ´ÿÀÝ. Ï gS(c) ∈ γ ÿÀÝaü:, Ù¥ c ∈ γ.
Ún 5.6.1 y²üÑ´Äu 5.6.2 ±9Ñy3·K 5.3.11 y²¥/IPE|.
Ún 5.6.1 y². ^/N ϕ = ϕ−1S(c2) ϕS(c1) : ∆S(c1) → ∆S(c2) O·K
5.3.11 ¥ φ : Ωc1 → Ωc2 , Ï B(S(c1)) = B(S(c2)), ¤± ϕ(S(c1)) = S(c2), ù´. Ï
§5.6. ½n 5.1.1 y² 93
∂∆S(ci) ´[, Ù¥ i = 1, 2, K ϕ ±òÿ[/N ϕ0 : C → C ¦ϕ0(∞) =∞ ϕ0 3 ∆S(c1) þ´ ϕ.
üå, ·ò gS(ci) Ú ∆S(ci) ¥eI S(ci) ^ i O, Ù¥ i = 1, 2. e5,
·^ gi ∈ Σλ ÄåXÚ5½Â[/N ϕn. ØÓu·K 5.3.11 y², ùp
òÿN´Ï·ØIÄ/X f−n1 (Ω1 \ f1(Ω1)) «. ·IÄ
Siegel n g_±9 C \ g−ni (∆i) ëÏ©|, Ù¥ i = 1, 2.
béz 1 ≤ k ≤ n−1,3[/N ϕk : C→ C¦ ϕk = g−12 ϕk−1g1
ϕk : g−k1 (∆1)→ g−k2 (∆2) ´X. y3·8B/½Â ϕn Xe.
é g−n1 (∆1) \ g−n+11 (∆1) ?Û©| ∆n
1 , ·o´Ué g−n2 (∆2) \ g−n+12 (∆2)
äkÓ/éA©| ∆n2 . y3·3 g−n+1
1 (∆1) þ½Â ϕn = ϕn−1 ½Â
ϕn = g−12 ϕn−1 g1 ¦§ò ∆n
1 N ∆n2 . Ï g−n1 (∆1) ´ëÏ, §Ö?Û©|
Ñ´üëÏ. aqu·K 5.3.11 ¥?Ø, ·±[/N ϕn : C→ C,
§3 g−n1 (∆1) ´X÷v ϕn = g−12 ϕn−1 g1.
aqu·K 5.3.11y²",·ü4[/N h1 Ú h2, §Ñ
± 0, π/2 Ú ∞ ØÄ. Ïé¤k n ≥ 1 k g2 ϕn = ϕn−1 g1, l h1 Ú h2 ÷v
g2 h1(z) = h2 g1(z). du3⋃n≥0 g
−n1 (∆1) þk h1(z) = h2(z), ùL²3E²¡
Èf8þk h1 = h2. dëY5, ùL² h1 = h2. u´ g1 Ú g2 ´*d[/Ý
. âÚn 5.6.2 1(Ø, g1 = g2.
Ún 5.6.3. 8Ü S(γ) ´ Σλ ¥^ Jordan , §d Σλ ¥¤k¦ u
: Siegel >.´[3Pê¿Âe²Lüéé¡.:¼ê g
|¤.
y². Ï S ´ëY, K B(S(c)) ëY6u c. Ïd, c l 0 ( u c = π/2)
π ( u c = −π/2) ,2 2π (£ c = π/2) Cz, B(S(c)) ´ëYCz.
âÚn 5.6.1, ·±½ÂN χ : [0, 2π] → S(γ), ¦éz β ∈ [0, 2π], k
χ(β) = S(cβ) Ú B(S(cβ)) = β. ùL² S(γ) ⊂ Σλ ´^ Jordan .
b c ∈ Σλ ¦ ∂∆gc ´²L ±π/2 Ú ±c [. P π/2 Ú c mÝ
B(S(c′)), aquÚn 5.6.1 y², gc Ú gS(c′) *d[/Ý. âÚn 5.6.2
1(Ø, · S(c′) = c. ùL² c ∈ S(γ).
- S(γ)int Ú S(γ)ext ©O Σλ \ S(γ) k.ÚÃ.©|. âíØ 5.5.12, S ´ëY, Ïd S(γint) ¹3 Σλ ¥. âÚn 5.6.3, ùL² S(γint) o¹3 S(γ)int ¥,
o¹3 S(γ)ext ¥.
·K 5.6.4. ÃâN S : Σ1/2 → Σλ ´÷.
94 1ÊÙ x¼êÄåXÚ
y². y²ù·K, ·y² S : γint ∪ γ → S(γ)int ∪ S(γ) Ú S : γext ∪ γ →S(γ)ext ∪ S(γ) Ñ´÷.
âc¡EÚ.:éA'X, S 7ò γint N\ S(γ)int. P γint =
γint ∪ γ ∪ f0 Ú S(γ)int = S(γ)int ∪ S(γ) ∪ g0, Ù¥ f0(z) = z/2 − z3/6 g0(z) =
e2πiθ(z − z3/3). ·äó S : γint → S(γ)int ´÷. ¯¢þ, 3üÓ h1 : γint → DÚ h2 : S(γint)→ D, ¦ h2 S h−1
1 : D→ D ´ëY, h2 S h−11 3ü ± S1
þ´ðN. XJÃâN S Ø´÷, K·òíÑ D kÂ Ø S1. ù
´ØU.
Ï γext dü©||¤, y²E,:. ·é γext ÑXed'
X ∼. é?¿ c1, c2 ∈ γext, c1 ¡ ∼ du c2, XJ (1) c1 = c2 ½ö (2) c1 = c2
|Re(c1)| = π/2. ·^ γ′ext 5L«ûm γext/ ∼. N´wÑ γ′ext ∪ f∞ Óuü . âcaq?Ø, k S : γ′ext → S(γ)′ext ÷. ùp γ′ext = γ′ext ∪ γ ∪ f∞ S(γ)′ext = S(γ)′ext ∪ S(γ) ∪ g∞, Ù¥ f∞(z) = sin(z)/2, g∞(z) = e2πiθ sin(z) S(γ)′ext
S(γ)ext ûm, ùûmþk γ′ext Ód'X ∼. ·Ky..
ã 5.2 ëêm Σλ 9Ù±Ï, Ù¥ λ = e2πiθ θ = (√
5− 1)/2 ^=ê.
ü4 S(γ) ⊂ [−π/2, π/2] × [−3, 3] çÚÚÚÜ©>., §©O
éAX S(γ)ext Ú S(γ)int. ã¡: [−4, 4]× [−3, 3].
üíØ, ·ke¡½n 5.6.5 Ú½n 5.6.6.
½n 5.6.5. - θ k..Ãnê. Ké?¿ c ∈ C \ kπ : k ∈ Z,
gc(z) = e2πiθ
(sin(z)− sin3(z)
3 sin2(c)
)
§5.6. ½n 5.1.1 y² 95
±:% Siegel >.´3Pê¿Âe²L 2 , 4 ½ö 6 fc
.:[.
½n 5.6.6. Jordan 4 S(γ) ⊂ Σλ ò Σλ ©¤üÜ© S(γ)int Ú S(γ)ext, ¦
(1) XJ c ∈ S(γ)int, K ∂∆gc TвL 2 .: c Ú −c.(2) XJ c ∈ S(γ)ext, K ∂∆gc TвL 2 .: π/2 Ú −π/2.
(3) XJ c ∈ S(γ) \ ±π/2, K ∂∆gc TвL 4 .: ±π/2 Ú ±c.(4) XJ c = π/2 ½ −π/2, K ∂∆gc TвL 2 ê 3 .: π/2 Ú −π/2.
½n 5.1.1 Ú½n 5.1.3 y². (ܽn 5.6.5, ½n 5.6.6 ±9Ún 5.2.3.
18Ù ng tableaux ¢y½n#y²
ùÙ¥, |^ Emerson [30], DeMarco Ú McMullen [18] Ú\äþÄåXÚ,
·Ñ Branner Ú Hubbard ng. tableau ¢y½n#y².
§6.1 Ä tableaux
ngõªÄåXÚ®²é\/ïÄ. Ù¥²;óáu Branner Ú
Hubbard [10, 11]. 3 [11] ¥, ¦Ú\^uïÄngõª Julia 8|Ü(
rkåóä: tableau (L). ¦ÓÄ tableaux ¢y¯K, =äkü
.: tableau oÿ±ngõª¤¢y.
éuäkEXêngõª f , 8Ü
K(f) = z ∈ C | S f n(z)n≥1 k.
¡ f W¿ Julia 8, Ù¥ f n L« f n gEÜ. ÙÖ Ω(f) := C \ K(f) Ã
¡áÚ, K(f) >.½Â f Julia 8 J(f). ®², <ºÝ¼ê
G : C→ [0,+∞) ½Â
G(z) = limn→∞
1
3nlog+ |f n(z)|.
§´ëY, ÷v G(f(z)) = 3G(z) Ú K(f) = G−1(0); [10, 53].
bngõª f Julia 8Ø´ëÏ, K3 f .:á3 Ω(f)
¥. - c0 Ú c1 f ü.:, c0 <º¯@, = G(c0) ≥ G(c1). P
r0 = G(c0) > 0. é?¿ k ≥ 0, 8Ü G−1([0, r0/3k−1)) kõpØmÿ
À¿. zùm¡´Ý k ©ã¡, P Pk. éz¦
G(c0)/3N ≤ G(c1) < G(c0)/3N−1 ê N ≥ 1, .©ã¡ Pk(c1) ½Â¹ c1
G−1([0, r0/3k−1)) ëÏ©|, Ù¥ 0 ≤ k < N + 1. ùp N #Nuá, =du
c1 Ø<º, Ï G(c1) = 0 d3áõ.©ã¡.
õª f N ng tableau ½IP: (marked grid) ´de^
½Â M(j, k) ∈ 0, 1 | j, k ≥ 0 j + k ≤ N:
M(j, k) = 1 = f k(c1) ∈ Pj(c1).
XJ M(j, k) = 1, K M(j, k) ¡´IP. IP:±w´1o
Z2 :, Ù¥ j ≥ 0 L«l x ¶ål, k ≥ 0 L«lK y ¶ål. Branner Ú
Hubbard [11]y²ngõg®IP:7L÷vK. N (Ä
96
§6.2. äþÄåXÚ 97
)IP: M = M(j, k) ∈ 0, 1 | j, k ≥ 0 j + k ≤ N ´÷ve^ [11, 19]:
(R0) éz n ≤ N , M(n, 0) = M(0, n) = 1.
(R1) XJ M(j, k) = 1, Ké¤k i ≤ j k M(i, k) = 1.
(R2) XJ M(j, k) = 1, Ké¤k 0 ≤ i ≤ j k M(j − i, k + i) = M(j − i, i).(R3) XJ j + k < N , M(j, k) = 1, M(j + 1, k) = 0, M(j − i, i) = 0 é 0 < i < m ¤
á, M(j −m+ 1,m) = 1, K M(j −m+ 1, k +m) = 0.
(R4) XJ j + k < N , M(j, k) = 1, M(1, j) = 0, M(j + 1, k) = 0, é¤k
0 < i < j k M(j − i, k + i) = 0, K M(1, j + k) = 1.
K (R4) 3 [11] ¥vkÑy, 3¦y²¥(¢^. K (R4) ©
ÄkÑy3©z [34] ¥, ´þ¡ãK (R4) «AÏ/. , Ù
(ãd Kiwi [42], DeMarco Ú McMullen [18] Õá/é.
P M N IP:, XJ f tableau M Ó, K¡ M ù
ngõª f ¤¢y. Branner Ú Hubbard y²e¡½n.
½n 6.1.1 (Tableaux¢y, [11]). ?ÛIP:ѱdngõª¤¢y.
5`, ¢y½IP:ngõªØ´. ùÙ·ò^ä
þÄåXÚѽn 6.1.1 #y². ÙÌg´: k`²?Û
N IP:±dÓ “Д ng children N¢y, ùng
children N±òÿNÝu 3 aõªä. duzngaõªä
ѱdngõª¤¢y, l ½n 6.1.1 y.
§6.2 äþÄåXÚ
P¹õª f 3 Ω(f) þ|Ü&E, Emerson [30], DeMarco Ú McMullen [18]
kÚ\äþÄåXÚ.
Äk, ·£Á3 [18] ¥½ÂÄäÄåXÚk'½Â. üXä T
´,ëÏ,ÛÜk,ÃüXE/. ä T º:Ú> (vk!4
) ¤8Ü©OP¤ V (T ) Ú E(T ). ½º: v ∈ V (T ) >/¤k
8 Ev(T ), ÙÄê½Âº: v val(v). äà¤m ∂T ´ØëÏ
m, §±dé T \K ëÏ©|¤8Ü_4, Ù¥ K H T ¤kk
fä.
F : T1 → T2 ´üüXämN, XJ (1) F ´_;, mëY
; (2) F ´üXN (T1 ¥z^>Ñ5/N T2 ¥,^>), K¡ F ´
98 18Ù ng tableaux ¢y½n#y²
©|CX. ©|CX F : T1 → T2 ÛÜNݼê´÷ve^N
deg : E(T1) ∪ V (T1)→ 1, 2, 3, · · · : ézv ∈ V (T1), k
2 deg(v)− 2 ≥∑
e∈Ev(T1)
(deg(e)− 1), (6.2.1)
±9éz e ∈ Ev(T1), k
deg(v) =∑
e′∈Ev(T1):F (e′)=F (e)
deg(e′). (6.2.2)
ÛNÝ deg(F ) ½Â
deg(F ) =∑
F (e1)=e2
deg(e1) =∑
F (v1)=v2
deg(v1), (6.2.3)
Ù¥ e2 Ú v2 T2 ¥?¿>Úº:. |^ (6.2.2) ±9 T2 ëÏ5, N´y (6.2.3)
½Â´Ün. ùÙ¥, ·=Ä deg(F ) = 3 /.
½Â 6.2.1. - F : T → T üXägC©|CXN, x, y ´ T þü
:. XJé, m,n > 0 k F m(x) = F n(y), K¡ x, y á3Ó;. XJe¡ 4
^¤á, K¡©|CXN F : T → T ´aõª.
(1) 3áà ∞ ∈ ∂T ;
(2) 3÷v (6.2.1) Ú (6.2.2) ÛÜNݼê;
(3) ä T vkf ( 1 º:);
(4) ?Ûº:;ѹu½u 3 º:.
d, | (T, F ) ¡´aõªä.
©z [18] ¥½n 2.9 L²: XJ©|CXN F : T → T k÷v (6.2.1)
Ú (6.2.2) ÛÜNݼê, @oùÛÜNݼê´. aõªk'
[&E±3 [18] ¥é.
- f ´ngõª Julia 8ëÏ, ò G Y²8zëÏ©|l¤:
ûm´üXä, P T . u´ f pûäþN F : T → T .
| τ(f) = (T, F ) ¡´ f û. d [18, ½n 3.1], f û τ(f) ´aõªä. ©z
[18] ¥Ì(J´e¡½n.
½n 6.2.2 (ä¢y, [18]). ?Ûaõªä (T, F ) Ñ´,õª f û.
T káàüXä (5¿ T fع3S). ?ÀÙ¥
áàIP ∞, Kz¿©C ∞ º: 2. XJ T ku½u 3
§6.2. äþÄåXÚ 99
º:, K73C ∞ u½u 3 º:, P v0, ·¡ùº
: T Ä:. XJ T zº:Ñu 3, ?ÀÙ¥u 2 º:, IP
v0.
|Üpݼê h : V (T ) → Z ½Â: |h(v)| ë v Ú v0 ¤I>
^ê. d, h ÎÒde¡^(½: XJ v á3ë v0 ∞ á´p, K
h(v) ≤ 0, ÄK h(v) > 0.
XJüº: v, w ÷v h(v) = h(w) + 1 k^>ë v Ú w, Kº: v ¡´
º: w child.
½Â 6.2.3. F : T → T ´üXN, XJézº: v ∈ V (T ), ÷v v
z child Ñ´ F (v) child, @o F ¡´± children .
5¿aõª½´± children . N´wÑé± children N
F : T → T 5`, XJ h(v) = k, K h(F (v)) = k + U(F ), Ù¥ U(F ) ´=6u F
~ê. d, aõªäØÓ´, ± children Nؽ´÷.
üXä T ©| (branch)´S (vk)nk=1, Ù¥ h(vk) = k vk+1 ´ vk
child, 1 ≤ k < n, ùpê n ¦´, = n =∞ ½ vn u 1.
½Â 6.2.4. XJ± children N F : T → T ÷ve^, K¡Ù´
N > 0 ng children N.
(a) U(F ) = −1;
(b) 3Nݼê deg : V (T ) ∪ E(T )→ 1, 2, 3 ÷v(b1) é,©| (sk)
Nk=1 k deg(sk) = 2, Ù¥ h(sk) = k 1 ≤ k < N + 1;
(b2) é l ≤ 0 k deg(sl) = 3, Ù¥ s0 = v0 Ä: h(sl) = l ≤ 0.
(b3) ézº: v ∈ T \ ∪k<N+1sk k deg(v) = 1.
(b4) - v1 Ú v2 > e ∈ E(T ) üà:. XJ h(v1) = h(v2) + 1, K deg(e) =
deg(v1).
Ï skk<0 3ä T þÄåXÚ¥¿Ø, S S := (sk)k<N+1 ¡´ng
children N F : T → T .©|.
½Â 6.2.4 ^ (a) L«3 F ^e, äXáà ∞ c?Ú.
5¿·=ÄÝ N ¢y¯K, Ï kü<º.: c0 Ú c1 ÷v
G(c0)/3N ≤ G(c1) < G(c0)/3N−1 ngõªU¢y N IP:.
3ù«¹e, dTngõªpaõªäkü<º.: (= U(F ) = −2).
, , ·±ÏLK τ(c1) 3Ýu N ;, Ó½ÂÐaõ
ªä, Ù¥ τ ò G Y²8ëÏ©|l¤ü:@ûN. ¯¢þ, éu?¿
100 18Ù ng tableaux ¢y½n#y²
N < ∞ IP:, Ñ3ngõª¦Ù¥.:Ø<º
Ó¢yù½IP:.
§6.3 ng children Nòÿ
éu N IP:, ·IEÓng childrenN
¦ÄåXÚ´½ÂÐ. Ó, ·7L`²dùIP:ùng±
children N±òÿNÝu 3 aõªä, l â½n 6.2.2 ½n
6.1.1 ¤á.
- F ′ : T ′ → T ′ Ú F : T → T üng children N. N F ′ : T ′ → T ′ ¡
´ F : T → T òÿ´: T ´ T ′ fä, F ′ 3 T þu F §3
T þkÓNݼê. - C(v) L« v ¤k children ¤8Ü.
Ún 6.3.1 (òÿÚn). - S = (sk)k<N+1 L« N ≥ 1ng children
N F : T → T .©|. K F : T → T ±òÿNÝu 3 aõª
ä=
(1) é v ∈ V (T ) \ S, XJ v1, v2 ∈ C(v) v1 6= v2, K F (v1) 6= F (v2);
(2) é i 6= 0, XJ u ∈ C(si) u 6= si+1, K F (u) 6= F (si+1);
(3) XJ si k 3 ØÓ children u, v, w F (u) = F (v), K F (w) 6= F (u);
(4) é?¿ v ∈ V (T ), k∑
F (u)=v deg(u) ≤ 3.
y². XJng children N F : T → T ±òÿngaõªä
F ′ : T ′ → T ′, K (6.2.1)-(6.2.3) éu F ′ : T ′ → T ′ ¤á. ±y (6.2.2) íÑ (1),
(2) Ú (3) (6.2.3) íÑ (4). ùy²75.
·ÏL8By²¿©5. Ä T Ä: s0 (= v0), d^ (4) · s0
õkü children. XJ C(s0) = s1,· s0 \þÝu 1 child¦ (6.2.3)
¤á. Pòÿng children N F1 : T1 → T1. Kª (6.2.2) éu T1 ¥@
|ÜpÝu 1 º:´¤á.
b (6.2.1)-(6.2.3) éu Tk ¥@|ÜpÝu k º:´¤á, Ù¥ k ≥ 1.
éz¦ h(v) = k º: v ∈ Tk, u v children. XJª (6.2.2) ؤá, v
\þvõÝu 1 children ¦ª (6.2.2) ¤á. y3·ng
children N Fk+1 : Tk+1 → Tk+1. ùN´ Fk : Tk → Tk òÿ, ¦ª (6.2.2)
é Tk+1 ¥@|ÜpÝu k + 1 º:´¤á.
^ù«ª, ·±ò F : T → T òÿ ∞ ng children N
F∞ : T∞ → T∞. d, ªf (6.2.1)-(6.2.3) éu T∞ ¥¤kº:Ú>Ѥá. 'aõ
§6.4. lIP:Ðng CHILDREN N 101
ªä¤I 4 ^, ¿5¿ÛNÝ deg(F∞) =∑
F∞(v)=s0deg(v) = 3,
F∞ : T∞ → T∞ ´NÝu 3 aõªä.
5 6.3.2. Ún 6.3.1 4 ^´ [18] ¥Ongõªäè'. éuõ
.: tableaux ¢y¯K, ^ (2), (3) Ú (4)Ø2´7.
½Â 6.3.3. XJng children N F : T → T ÷vÚn 6.3.1 ¥^, K
¡§´Ð (nice).
§6.4 lIP:Ðng children N
ù!¥, ·F"l½½IP:¥JÐng chil-
dren N. ·¡ N IP: M ÓÐng children
N F : T → T ¤¢y, XJ§÷vXe^:
M(j, k) = 1 = deg(F k(sj+k)) ≥ 2.
Ún 6.4.1. z N IP:þÓÐng children
N¤¢y.
y². ·|^8By²ùÚn. éu N = 1, - T1 = (sk)−∞k=1 ¦ sk ´ sk−1
child. ½Â F1 : T1 → T1 F1(sk) = sk−1, Ù¥ k ≤ 1. Nݼê½Â
deg(s1) = 2 deg(si) = 3, Ù¥ i ≤ 0. w,, F1 : T1 → T1 ´Ðng children
N, ¿¢y@ 1 IP:.
XJ N = 2, KküØÓ 2IP:, ¦©OéAX M(1, 1)
IP½vIP. XJ M(1, 1) IP, s1 \þÝ 2 child s2, ·
üXä T2.
ò F1 : T1 → T1 òÿ F2 : T2 → T2 ¦éu¤k k ≤ 2 k F2(sk) = sk−1, K
F2 : T2 → T2 ´Ðng childrenN¿Ó¢y 2M(1, 1) = 1I
P:. aq/,XJ M(1, 1)vkIP, s1 \þÝ 2child s2 s0 \þ
Ý 1 child v1, ·üXä T ′2. y3ò F1 : T1 → T1 òÿ F ′2 : T ′2 → T ′2
¿¦ F ′2(s2) = v1 F ′2(v1) = s0. K F ′2 : T ′2 → T ′2 ´Ðng children N
Ó¢y 2 M(1, 1) = 0 IP:.
b?Û k ≤ N − 1 IP:þÓÐng children
N Fk : Tk → Tk ¤¢y, Ù¥ N < +∞. â8BL§, üXä Tk á3.©|
(si)i≤k c; (forward orbit) ¥. - M ½ N IP:, e
5?Øòl M(N, 0) Ñu, 3IP:¥÷XÀ M(0, N) (å.
102 18Ù ng tableaux ¢y½n#y²
Äk, sN−1 \þÝ 2 child sN ,,£ M(N−1, 1). XJM(N−1, 1) IP, KâK (R2) N :ÑIP. ½Â FN(sN) =
sN−1,·Ðng childrenN FN : TN → TN ,TN¢y M(N−1, 1) IP: M .
b M(N − 1, 1) vkIP. FN−1(sN−1) \þÝ 1 child vN−1
½Â FN(sN) = vN−1. ,£ M(N − 2, 2). XJ M(N − 2, 2) IP, ½Â
FN(vN−1) = sN−2. N´y FN : TN → TN ´Ðng children N¢y
M(N − 1, 1) vkIP M(N − 2, 2) IP:.
b M(N − 2, 2) vkIP. éu M ¥?Û M(m,n), P M(m,n) þ
n/ Q(m,n) = (M(j, k)) | j + k ≤ m + n, 0 ≤ j ≤ m n ≤ k ≤ m + n.XJ Q(N − 2, 2) 6= Q(N − 2, 1), F 2N−1(sN−1) \þÝ 1 child vN−2, ½Â
FN(vN−1) = vN−2. ,£ M(N − 3, 3). XJ Q(N − 2, 2) = Q(N − 2, 1), K
M(N − 2, 1) vkIP. ½Â FN(vN−1) = FN−1(sN−1), Ún 6.3.1 ¥ 4 ^ÑN
´y÷v.
bXéuN FN ·®²3; sN 7→ FN(sN) 7→ · · · 7→ F kN (sN) þ½Â, Ù
¥ k ≥ 1 (5¿ FN 3 F kN (sN) þvk½Â), vN−i = F iN (sN) ´ F iN−1(sN−1) #
\Ý 1 child, ùp 1 ≤ i ≤ k. d, éu 1 ≤ i ≤ k k M(N − i, i) = 0, ù´Ï
XJ·-IP @o8BEÒ(å.
y3£ M(N − k − 1, k + 1) þ, ·^Úc¡aq?Ø. XJ M(N − k −1, k+ 1) IP, ½Â FN(vN−k) = sN−k−1. 3ù«¹e, ·IyÚn 6.3.1 ¥
^ (4). ¯¢þ, y3, Ø vN−k, âE, sN−k−1 vkÙ§Ý 1 _
. ùL²∑
F (u)=sN−k−1deg(u) ≤ 3.
P43E¥,·o\\ childrenÝÑu 1oâK (R2),E
(å. XJ M(N −k−1, k+ 1)vkIP,u Q(N −k−1, k+ 1)Ú Q(N −k−1, j),
Ù¥ 0 < j ≤ k. XJé?¿ 0 < j ≤ k, k Q(N − k − 1, k + 1) 6= Q(N − k − 1, j),
F(k+1)N−1 (sN−1) \þ child vN−k−1 ¿½Â FN(vN−k) = vN−k−1. ,·£
M(N − k − 2, k + 2). XJ3, j ¦ Q(N − k − 1, k + 1) = Q(N − k − 1, j), @o
·`²¯. XJ M(N − k − 1, k), M(N − k − 1, j − 1) M(N − k, j − 1)
þIP, F(k+1)N−1 (sN−1) \þ child vN−k−1 ¿½Â FN(vN−k) = vN−k−1. ,
·£ M(N − k− 2, k+ 2). ÄK, ·½ÂFN(vN−k) = F jN−k−j−1(sN−k−j−1). a
q/, â tableaux KÚ·EÚn 6.3.1 ¥ 4 ^þ¤á.
^ù, ·o´±Ðng children N¿¢y½
N IP:. d8B, ½ny².
5 6.4.2. ·éÚn 6.4.1y²¥ M(N − 2, 2) = 0Ú Q(N − 2, 2) = Q(N − 2, 1)
§6.4. lIP:Ðng CHILDREN N 103
/Ñ5º. ¯¢þ, ÷vÚn 6.3.1 ¥^ (2), ·Iu M(N −2, 1), M(N − 2, 0) Ú M(N − 1, 1). Ï M(N − 2, 1) = 0, ·½Â FN(vN−1) =
FN−1(sN−1). , , XJ´/ M(N − k − 1, k + 1) = 0 é, 0 < j ≤ k k
Q(N − k − 1, k + 1) = Q(N − k − 1, j), K·IaquÚn 6.4.1 ¥y².
½IP:±ü½õÐng children N¢y (ã 6.1 ±
9 §6.5). N´yéu N ≤ 4, N IP:UÓÐ
ng children N¢y.
s−1
s0
s1
s2
s3
s4
s5
s−1
s0
s1
s2
s3
s4
s5
0
1
2
3
4
5
0 1 2 3 4 5
ã 6.1 N = 5 IP:, §±düÐng children N
¢y. ¥mä´dÚn 6.4.1 ¥E. ¢%:L.
%:KL. .
½n 6.1.1 y². (ܽn 6.2.2 ±9Ún 6.3.1 Ú 6.4.1 (J.
N 1 2 3 4 5 6 7
MGN 1 2 4 8 16 33 69
NCN 1 2 4 8 18 42 103
N 8 9 10 11 12 13 14
MGN 144 303 641 1361 2895 6174 13188
NCN 260 670 1753 4644 12433 33581 91399
N 15 16 17 18 19 20 21
MGN 28229 60515 129940 279415 601742 1297671 2802318
NCN 250452 690429 1913501 5328648 14902959 41841737 117887513
L 6.1 1 ≤ N ≤ 21 , IP:ÚÐng children Nê8. ÎÒ
MGN Ú NCN ©OL«IP:ÚÐng children Nê8.
104 18Ù ng tableaux ¢y½n#y²
§6.5 IP:ÚÐng children Nê8
âc¡ãK (R0)-(R4), ·±O½ØÓIP:ê8. a
q/, dÚn 6.3.1 ¥ã 4 ^, ·±O½ØÓÐng
children Nê8 (L 6.1).
5¿3©z [18] ¥, ØÓÐng children Nê8 ([18] ¥¡´ä
ê8) ®²O N ≤ 17. ùp·ò(Jí2 N ≤ 21.
1ÔÙ McMullen NááÚ>.
Hausdorff ê
§7.1 ½nã
éuê Q ≥ 3, McMullen N
fp(z) = zQ + p/zQ
ÄåXÚ®²þïÄ ([20, 22, 24, 69, 61]). p é, ùAÏkn¼
ê±w¤´ f0(z) = zQ ü6Ä. 5¿ùp·ò (4.1.1) ¥ëê λ ¤ p Ì
´ [75] ¥±.
l [20, 49] ®²: éu p, fp Julia 8´ Cantor ±. 3ù«¹
e, ¤k Fatou ©|Ñ ∞ ¤3 Fatou ©|¤áÚ. ·^ Bp L« ∞ áÚ,
K>. ∂Bp ´^ Jordan (âÚn 7.2.3 ¯¢þ´[). ¯¢þ, [61]
¥y² ∂Bp XJ J(fp) Ø´ Cantor 8, K>. ∂Bp o´^ Jordan . 3ù
Ù¥, ·y²e¡½n.
½n 7.1.1. - Q ≥ 3, Kéu¦ J(fp)´ Cantor± p, ∂Bp Hausdorff
ê´
dimH(∂Bp) = 1 +|p|2
logQ+O(|p|3). (7.1.1)
AO/, XJ Q 6= 4, Kp O(|p|3) ±d O(|p|4) O.
íØ, p é, þã½nÑ J(fp) Hausdorff ê
e..
éuõª Pc(z) = zd + c, Ù¥ d ≥ 2 c é¦ Pc ´V, Pc Julia 8
Hausdorff ꮲ3 [65], [75] Ú [15] ¥©OO, Ùêúª©OÐm'
u c 1, 1nÚ1o. nØþ, pѱëY/¦Ñ5. , , X,
p, O¬5E,.
§7.2 úªí
½n 7.1.1 y²aqu [54] Ú [75]. ¤k¡Oò33e!. e5,
·o´b½ p v (p = 0 ´#N).
105
106 1ÔÙ McMullen NááÚ>. Hausdorff ê
ã 7.1 fp(z) = zQ + p/zQ Julia 8, Ù¥ p = 0.005 Q = 3, 4. üÑ´
Cantor ±. ã¡: [−1.25, 1.25]× [−1.25, 1.25].
Øõª Pc(z) = zd + c, McMullenxëêmkÛ: p = 0. pªu
0 , Julia 8 J(fp) 3 Hausdorff ÿÀeØÂñ J(f0) (ü ± S1), [22].
, , ááÚ>. ∂Bp %´Âñ. ¯¢þ, ·±`² (Ún 7.2.3)
∂Bp ´ü ± S1 X$Ä. d, ·Äk£X$ĽÂ.
½Â 7.2.1 (X$Ä, [48]). - E E¥¡ C f8, N h : D×E → C ¡´ E ± D ëêm, 0 Ä:X$Ä, XJTN÷v
(1) éz z ∈ E, β 7→ h(β, z) 'u β ∈ D ´X;
(2) éz β ∈ D, z 7→ h(β, z) 3 E þ´ü;
(3) h(0, z) = z é¤k z ∈ E ¤á.
½Â 7.2.1 ¥ü D ±dÙ§ÿÀO.
½n 7.2.2 (λ Ún, [48]). E X$Ä h : D×E → C ±/òÿ E
X$Ä h : D × E → C. éz β ∈ D, N h(β, ·) : E → C ±òÿE¥¡g[/N.
l [69] ®²: McMullen «M := p ∈ C− 0; J(fp) Cantor ± ´:%. ù`² V =M∪ 0 ¹ 0 ÿÀ.
Ún 7.2.3. 3X$Ä H : V × S1 → C ± V ëêm± 0 Ä:¦é
u¤k p ∈ V k H(p,S1) = ∂Bp.
y². ·Äky² f0(z) = zQ z½5±Ï:3 V ¥´X/$Ä. -
z0 ∈ S1 ù±Ï k ±Ï:. éu p, N fp ´ f0 6Ä.
§7.2. úªí 107
âÛ¼ê½n, 3 0 U0 ¦éu¤k p ∈ U0, z0 ¤ fp ½5±
Ï: zp, äkÓ±Ï k. ,¡, éu¤k p ∈ M, Ï fp vk½5±Ï
;, fp z½5±Ï;3M ¥Ñ´X/$Ä ( [50] ¥½n 4.2).
Ï V ´üëÏ, éu p ∈ U0, 3XN Z : V → C ¦ Z(p) = zp. -
Fix(f0) f0 ¤k½5±Ï:. K½Â h(p, z0) = Z(p) N h : V × Fix(f0)→ C´X$Ä. 5¿ S1 = Fix(f0), â½n 7.2.2, 3 h òÿ, P H :
V × S1 → C. w, H(p,S1) ´ J(fp) ëÏ©|.
e5, ·`²éu¤k p ∈ V , H(p, S1) = ∂Bp. âV Julia 8X$Ä
5, Iy²éuëê p ∈ (0, ε) k H(p,S1) = ∂Bp ¤á, Ù¥ ε > 0.
Ä f0 fp 6Ä, Ù¥ëê p ∈ (0, ε) é, f0 ØÄ: z0 = 1 ¤ fp
½5ØÄ: zp, ù½5ØÄ:´¢C 1. N fp kü¢ØÄ:. Ù¥
´ zp ,´C 0 z∗p . w, zp fp Ý 0 Xº:. Ïd
H(p, 1) = zp ∈ ∂Bp. ùL²éu¤k p ∈ (0, ε) k H(p,S1) = ∂Bp.
½Â 7.2.4 ([65]). - M ê N ¢)Û6/, J ´ M ;f8,
V ´ J 3 M ¥m. ¡ J ´¢)ÛN f : V →M ½f (repeller) X
Je¡n^¤á:
(1) 3~ê C > 0, α > 1 ¦ ‖(Txf n)u‖ ≥ Cα‖u‖ é?¿ x ∈ J, u ∈ TxM Ún ≥ 1 ¤á;
(2) J = x ∈ V :é¤k n > 0 k f nx ∈ V ;(3) é?¿ J m8 O, 3, n > 0 ¦ J ⊂ f n(O).
â½Â 7.2.4, >. ∂Bp N fp ½f.
½n 7.2.5 ([65,íØ 5]). XJx¢)ÛN fλ ½f Jλ )Û6uëê λ (=
(λ, x)→ fλ(x) ´¢)ÛN), @o Jλ Hausdorff ê¢)Û6u λ.
·½Â¼ê H : V → R+ H(p) = dimH(∂Bp). e5·ÄkéÑ H
Ä5. du ∂B0 = S1, K H(0) = 1. â Ruelle ½n, · H ´¢
)Û¼ê. Ïd p 3 0 NC, ·k
H(p) =∑s,t≥0
astpspt, a00 = 1. (7.2.1)
â fp(z) = fp(z) Ú f 2e2πi/(Q−1)p
(eπi/(Q−1)z) = eπi/(Q−1)f 2p (z) k
H(p) = H(p) = H(p), H(e2πi/(Q−1)p) = H(p).
108 1ÔÙ McMullen NááÚ>. Hausdorff ê
ÏdXê÷v
ast = ast = ats, ast = aste2πi(s−t)/(Q−1).
AO/, XJ s− t 6= 0 mod (Q− 1), K ast = 0. Ïd·k
H(p) =
1 + a20(p2 + p2) + a11|p|2 +O(|p|4), if Q = 3,
1 + a11|p|2 +O(|p|3), if Q = 4,
1 + a11|p|2 +O(|p|4), if Q ≥ 5.
O ∂Bp Hausdorff ê, ·Ie¡(J ( [31], ½n 9.1, ·K 9.6 Ú
9.7).
½n 7.2.6 (Falconer, [31]). - S1, · · · , Sm Rn 4f8 D þØ N¦
|Si(x)− Si(y)| ≤ ci|x− y|, Ù¥ ci < 1. K
(1) 3;8 J ¦ J =⋃m
i=1 Si(J).
(2) J Hausdorff ê H(J) ÷v H(J) ≤ s, Ù¥∑m
i=1 csi = 1.
(3) XJ?Ú¦ |Si(x)− Si(y)| ≥ bi|x− y|, Ù¥ i = 1, · · · ,m, K H(J) ≥ s, Ù
¥∑m
i=1 bsi = 1.
y3, ·k
Ún 7.2.7. é?Û p ∈ V, ∂Bp Hausdorff ê D = H(p) de¡ªû½:∑z∈Fix(fnp )∩∂Bp
|(f np )′(z)|−D = O(1). (7.2.2)
y². - wp ∈ ∂Bp fp Ý 0 Xº:. ·±ò wp w¤ü:
w+p Ú w−p ò ∂Bp w¤´äkà: w+
p Ú w−p 4ã. N f np : ∂Bp → ∂Bp
k Qn _©|, P S1, · · · , SQn , zÑò ∂Bp N4ã, ¦§U_
ü. d, ∂Bp =⋃Si(∂Bp). AO/, S1(∂Bp) Ú SQn(∂Bp) ¹à: wp. éu
1 < j < Qn, Sj(∂Bp) =¹ f np ØÄ:. â Koebe ½n¿5¿ ∂Bp ´
[, K3ü n Ã'~ê C1, C2, ¦
C1
|(f np )′(ζ)|≤ |Si(x)− Si(y)|
|x− y|≤ C2
|(f np )′(ζ)|, ∀ 1 ≤ i ≤ Qn, x, y ∈ Si(∂Bp),
Ù¥ ζ f np á3 Si(∂Bp) ¥ØÄ:.
â½n 7.2.6, ·k s1 ≤ D ≤ s2, Ù¥∑
ζ Csjj |(f np )′(ζ)|−sj = 1, j = 1, 2. ùL
² n éÿ, Ú∑
ζ |(f np )′(ζ)|−D ´0u C−D1 Ú C−D2 mê∑z∈Fix(fnp )∩∂Bp
|(f np )′(z)|−D =∑ζ
|(f np )′(ζ)|−D − |(f np )′(wp)|−D = O(1).
§7.2. úªí 109
Úny..
½n 7.1.1 y². 5¿ p = 0 , Julia 8 J(fp) ü ±±ëêz
z(t) = e2πit ¿¦
fp(z(t)) = z(Qt). (7.2.3)
éu p 6= 0, fp : ∂Bp → ∂Bp ´NÝ Q CXN. Ï ∂Bp Óuü
±, l ∂Bp ±ëêz¦ (7.2.3)¤á. âÚn 7.2.3, · ∂Bp þ: z(t)
´X/$Ä. ùL², 3 0 S, ·±ò z(t) Ðm
z(t) = e2πit(1 + pU1(t) + p2 U2(t) +O(p3)), (7.2.4)
Ù¥ Um(t) éu m ≥ 1 ÷v Um(t + 1) = Um(t). ò (7.2.4) \ (7.2.3), ,' p
k'ÐmXê, ·ke¡ª
U1(Qt)−QU1(t) = e−2πi(2Q)t, (7.2.5)
U2(Qt)−QU2(t) =Q(Q− 1)
2U2
1 (t)− e−2πi(2Q)tQU1(t). (7.2.6)
N´y5¼ê§ φ(Qt)−Qφ(t) = e−2πit k)
φ(t) = − 1
Q
∞∑l=0
Q−le−2πiQlt. (7.2.7)
Ï ·±)ѧ (7.2.5) Ú (7.2.6)
U1(t) = φ(2Qt), (7.2.8)
U2(t) =Q(Q− 1)
2
∞∑l1, l2=1
Q−(l1+l2)φ(2(Ql1 +Ql2)t) +Q
∞∑l=1
Q−lφ(2(Ql +Q)t).(7.2.9)
¯¢þ, p Um(t) ±ÏL8BOÑ5, Ù¥m ≥ 3. ùò¬~E,.
5¿ f np 3 ∂Bp þØÄ:
Fix(f np ) ∩ ∂Bp = z(tj) : tj = j/(Qn − 1), j = 0, 1, · · · , Qn − 2. (7.2.10)
d [75], ·Ú\²þÎÒ
〈G(t)〉n =1
Qn − 1
Qn−2∑j=0
G(tj). (7.2.11)
ù²þÎÒk'k^ª´
〈e2πimt〉n =
1 XJ m ≡ 0 mod Qn − 1,
0 Ù§.(7.2.12)
110 1ÔÙ McMullen NááÚ>. Hausdorff ê
du
(f np )′(z(tj)) =n−1∏m=0
f ′p(z(Qmtj)) = Qn
n−1∏m=0
(zQ−1(Qmtj)−
p
zQ+1(Qmtj)
), (7.2.13)
·±ò (7.2.2)
O(1) = Q−nD(Qn − 1)
⟨n−1∏m=0
∣∣∣∣zQ−1(Qmt)− p
zQ+1(Qmt)
∣∣∣∣−D⟩n
. (7.2.14)
N¹¥OL²éu¤k¿© n, ·k
O(1) = Q−nD(Qn − 1)(1 +D2n|p|2 +O(np3)). (7.2.15)
½, n, p vÿ, ·k
O(1) = exp(n(−(D − 1) logQ+D2|p|2)
). (7.2.16)
ùL²
D = 1 +|p|2
logQ+O(|p|3), (7.2.17)
ù´½n 7.1.1 ¥Iúª.
§7.3 ON¹
ù!òÌy²ªf (7.2.15). Äk,·éÎÒz. ·©O^ zm, U1,m
Ú U2,m 5L« z(Qmt), U1(Qmt) Ú U2(Qmt). - σ = e2πit, â (7.2.4), ·k∣∣zQ−1m − p/zQ+1
m
∣∣ =∣∣∣(1 + Vm)Q−1 − σ−2Qm+1
p/(1 + Vm)Q+1∣∣∣
=∣∣∣1 + (Q− 1)Vm + (Q− 1)(Q− 2)V 2
m/2− σ−2Qm+1
p (1− (Q+ 1)Vm) +O(p3)∣∣∣ ,(7.3.1)
Ù¥ Vm = U1,m p+ U2,m p2 +O(p3). Ïd∣∣zQ−1
m − p/zQ+1m
∣∣−D2 =∣∣∣1 +
[(Q− 1)U1,m − σ−2Qm+1
]p
+[(Q− 1)(Q− 2)U2
1,m/2 + (Q+ 1)σ−2Qm+1
U1,m + (Q− 1)U2,m
]p2 +O(p3)
∣∣∣−D2=
∣∣∣∣1− D
2Amp+
D
8Bmp
2 +O(p3)
∣∣∣∣ ,(7.3.2)
§7.3. ON¹ 111
Ù¥
Am = (Q− 1)U1,m − σ−2Qm+1
, (7.3.3)
Bm = (Q− 1) (D(Q− 1) + 2)U21,m − 2(D(Q− 1) + 4Q)σ−2Qm+1
U1,m (7.3.4)
−4(Q− 1)U2,m + (D + 2)σ−4Qm+1
.
,·k∣∣zQ−1m − p/zQ+1
m
∣∣−D =
(1− D
2Amp+
D
8Bmp
2
)(1− D
2Amp+
D
8Bmp
2
)+O(p3)
= 1− D
2(Amp+ Amp) +
D2
4|p|2AmAm +
D
8(Bmp
2 +Bmp2) +O(p3).
(7.3.5)
Ún 7.3.1. - u, v ∈ N, é?Û n, K (1) 2Qv/(Qn − 1) 6≡ 0 mod 1; (2)
2Qv(Qu + 1)/(Qn − 1) 6≡ 0 mod 1.
y². Ï (Q,Qn − 1) = 1, ùL² (Qv, Qn − 1) = 1. qÏ n é 0 < 2 <
Qn − 1, l 2Qv/(Qn − 1) ØU´ê.
éu1(Ø, b u = ns+ r, Ù¥ 0 ≤ r < n, K
2Qv(Qu + 1)
Qn − 1≡ 2Qv Q
ns(Qr − 1) + 2
Qn − 1≡ 2Qv(Qr + 1)
Qn − 1mod 1.
Ï (Qv, Qn−1) = 1 né 0 < 2(Qr+1) < Qn−1,ùL² 2Qv(Qu+1)/(Qn−1)
ØU´ê.
âÚn 7.3.1, (ܲþ5 (7.2.12), N´y
íØ 7.3.2. 〈Am〉n = 0, 〈AmAk〉n = 0 〈Bm〉n = 0, Ù¥ 0 ≤ m, k ≤ n− 1.
â (7.3.5), ·k⟨n−1∏m=0
∣∣zQ−1m − p/zQ+1
m
∣∣−D⟩n
= 1 +D2
4|p|2
n−1∑m, k=0
〈AmAk〉n +O(|p|3). (7.3.6)
ò (7.3.3) \ AmAk, ·k
〈AmAk〉n = (Q− 1)2〈U1,mU1,k〉n + 〈σ2Q(Qk−Qm)〉n− (Q− 1) (〈σ−2Qm+1
U1,k〉n + 〈σ2Qk+1
U1,m〉n).(7.3.7)
Ún 7.3.3. - u ∈ N, K (Qu − 1)/(Qn − 1) ´ê= u = ns é,
s ∈ N ¤á.
112 1ÔÙ McMullen NááÚ>. Hausdorff ê
y². ¿©5´w,, ·=y75. b u = ns + r, Ù¥ 0 ≤ r < n, âb
·k
Qu − 1
Qn − 1≡ Qns(Qr − 1)
Qn − 1mod 1.
Ï (Qns, Qn−1) = 1,·Ñ (Qu−1)/(Qn−1)´ê= (Qr−1)/(Qn−1)
´ê, = r = 0.
âÚn 7.3.3 Ú²þÎÒ5 (7.2.12), k
n−1∑m, k=0
〈U1,mU1,k〉n =1
Q2
n−1∑m, k=0
∞∑l1, l2=0
1
Q l1+l2
⟨σ−2Q(Ql1+m−Ql2+k)
⟩n
=1
Q2
n−1∑m, k=0
( ∑l1+m=l2+k
1
Q l1+l2+∑v 6=0
∑l1+m=l2+k+nv
1
Q l1+l2
)
=1
Q2
(n−1∑m=0
m∑k=0
∞∑l1=0
1
Q 2l1+m−k +n−1∑k=0
k−1∑m=0
∞∑l2=0
1
Q 2l2+k−m
)
+1
Q2
(+∞∑v=1
n−1∑m, k=0
∞∑l2=0
1
Q 2l2+k−m+nv+−∞∑v=−1
n−1∑m, k=0
∞∑l1=0
1
Q 2l1+m−k−nv
)
=1
Q2 − 1
(Q+ 1
Q− 1n+O(1)
)+O(1) =
n
(Q− 1)2+O(1).
(7.3.8)
þ¡ª^e¡üª
n−1∑m=0
m∑k=0
1
Qm−k =nQ
Q− 1− Q−Q−(n−1)
(Q− 1)2=
nQ
Q− 1+O(1), (7.3.9)
n−1∑m=0
m−1∑k=0
1
Qm−k =n
Q− 1− Q−Q−(n−1)
(Q− 1)2=
n
Q− 1+O(1). (7.3.10)
(7.3.8) OL², n ªu ∞ , ¹ l1 +m 6= l2 + k éAÚk=6
u Q þ., ·^ O(1) L«. ù*:3e¡aqO¥´. =, (J¥
ÌÜ©Ñ´5g l1 +m = l2 + k ù«¹.
aqu (7.3.8) ¥O, ·k
n−1∑m, k=0
⟨σ2Q(Qk−Qm)
⟩n
= n+O(1) (7.3.11)
§7.3. ON¹ 113
n−1∑m, k=0
(⟨σ−2Qm+1
U1,k
⟩n
+⟨σ2Qk+1
U1,m
⟩n
)= − 2
Q
n−1∑m, k=0
∞∑l=0
1
Q l
⟨σ−2Q(Qm−Qk+l)
⟩n
= − 2
Q
n−1∑m=0
m∑k=0
1
Qm−k +O(1) = − 2n
Q− 1+O(1).
(7.3.12)
(Ü (7.3.7), (7.3.8), (7.3.11) Ú (7.3.12), ·k
n−1∑m, k=0
〈AmAk〉n = n+ n+ 2n+O(1) = 4n+O(1). (7.3.13)
â (7.3.6), ùL²⟨n−1∏m=0
∣∣zQ−1m − p/zQ+1
m
∣∣−D⟩n
= 1 +D2n|p|2 +O(np3). (7.3.14)
ª (7.2.15) y².
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®uLÚ®¤Ø©
1©Yang Fei and Yin Yongcheng, A new proof of the realization of cubic tableaux,Bull. Aust. Math. Soc., 87 (2013), 207-215.
2©Yang Fei, On the dynamics of e2πiθ sin(z) + α sin3(z). Accepted by Acta Math.Sinica, English Series.
3©Qiu Weiyuan, Yang Fei and Yin Yongcheng, Rational maps whose Julia sets areCantor circles, Arxiv 1301.2692v1. Accepted by Ergodic Theory and Dynamical Systems.
4©Yang Fei and Wang Xiaoguang, The Hausdorff dimension of the boundary of theimmediate basin of infinity of McMullen maps, Arxiv 1204.1282v1. Submitted.
5©Qiu Weiyuan, Yang Fei and Yin Yongcheng, A geometric characterization of theJulia sets of McMullen maps. Preprint, 2013.
119
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