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1/98 c/CIRCIDaeDuepes-p.“

METRIC PROPERTIES OF SEPARATED
SEMIHYPERBOLIC DYNAMICS WITH
APPLICATIONS TO HARMONIC MEASURE

By

Zoltan Balogh

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1995

ABSTRACT

METRIC PROPERTIES OF SEPARATED SEMIHYPERBOLIC DYNAMICS
WITH APPLICATIONS TO HARMONIC MEASURE

BY

Zoltan Balogh

Let f : V ——> U be a (generalized) polynomial-like map (GPL). Suppose that
harmonic measure w = w(-, 00) on the Julia set J I is equal to measure of maximal
entropy m for f : J j (—3. Then the dynamics (f, V, U) is called maximal. We are going
to prove that the condition w z m is sufficient for a fairly general class of dynamics

to be conformally equivalent to a maximal one, that is to be conformally maximal.

We introduce the class of separated semihyperbolic GPL. It turns out that sep-
arated semihyperbolic Julia sets have good self—similarity properties. Which gives
that the Jacobian of the harmonic measure has a Holder continuous logarithm. We
develop a version of thermodynamic formalism to construct a unique finite, invariant,
ergodic measure M such that ,u z w for the case of semihyperbolic f and totally dis-
connected J .Finally, the ”invariant harmonic measure”, p is being used to prove our
main result: Let (f, U, V) be a semihyperbolic GPL with totally disconnected Julia

set. Then w z m => (f, U, V) is conformally maximal.

DEDICATION

To Zsuzsanna

iii

ACKNOWLEDGEMENTS

I would like to express my deep gratitude to my teacher Alexander Volberg who
introduced me to the wonderful world of complex dynamics. During the last three
years I have experienced his continuous guidance and encouragement. In the process
of writing my dissertation I have received many helpful hints and even ideas from

Professor Volberg. I’m grateful for his generosity in sharing these with me.

While working on my dissertation I had the opportunity to meet a number of

mathematicians. Many interesting and stimulating conversations took place.

I would like to thank J uha Heinonen, Steffen Rohde, Mario Bonk, Jose Fernandez,
Kari Astala, John Smillie, Chris Bishop and Michael Lyubich for their interest and

helpful suggestions concerning the results in the first chapter.

I’m grateful to Martin Reimann for the invitation to the 3—rd Analysis Colloquium
in Bern, in August 1994. I would like to thank Michael Lyubich for inviting me to
the Complex Dynamics and Conformal Geometry Workshop at Berkeley MSRI in
May 1995. I also thank Michel Zinsmeister for the invitation to the Spring School in

Complex Analysis, in Seillac, May 1995.

I’m grateful to Bodil Branner, Alexander Eremenko, Joel Shapiro and Irina Popovici

for illuminating discussions on various problems in complex dynamics.

iv

Conversations with Mariusz Urbanski and Michel Zinsmeister were of great help

in understanding the ergodic theory of rational maps.

I’m especially thankful to Sheldon Newhouse and Manfred Denker for their kind

explanations on general ergodic theory.

Finally I would like to thank my wife Zsuzsanna and daughter Krisztina for their

love, understanding and support.

Contents

Introduction 1
0.1 Three measures in the plane ........................ 1
0.2 Hausdorff measure versus measure of maximal entropy ........ 2
0.3 Hausdorff measure versus harmonic measure .............. 4
0.4 Measure of maximal entropy versus harmonic measure ........ 6
0.5 The first question ............................. 6
0.6 The second question ........................... 8
0.7 Outline of main results .......................... 9

1 Geometric localization, uniformly John property and separated semi-
hyperbolic dynamics 15
1.1 Localization of simply connected John ................. 17
1.2 Localization for arbitrary John domains at a fixed scale ........ 24
1.3 Geometric properties of the Julia set .................. 28
1.4 Localization of A00( f) .......................... 33
1.5 Uniformly John domains ......................... 46

vi

Introduction

0.1 Three measures in the plane.

There are various ways to investigate the structure and properties of certain com-
plicated sets in the plane. One of the possible ways is by the harmonic measure
supported on the set. Another way would be by considering a geometric measure:

the Hausdorff measure.

The most sophisticated (and beautiful) sets appear often as a result of some
process of iteration ; and thus as invariant sets of some dynamical systems. In this
situation the ergodic thory is relevant; in particular it is meaningful to consider the

measure of maximal entropy.

So there are the three measures: harmonic measure - expressing complex analytical
properties ; the Hausdorff measure - reflecting geometrical content, and the measure
of maximal entropy - giving information about the dynamics. It is of general interest

to see the relations between these objects describing so different features.

A general idea in this respect is that these objects generally are different and
there are a small number of particular cases when two of the above three measures
are essentially the same. The problem of general interest is to describe these particular
cases. In what follows I will briefly recollect some of the recent achievments regarding

this circle of problems.

2

0.2 Hausdorff measure versus measure of maximal
entropy

Let f : C ——> C be a rational map on the Riemann sphere, deg( f) 2 2. A natural

invariant measure - the measure of maximal entropy, was constructed by Lyubich in

[Lyl],[Ly2] and independently by Freire, Lopes and Mafié in [FLM].

Let us denote the measure of maximal entropy by m, the Julia set of f by J and
the Hausdorff dimension of J is denoted by dim(J). The Hausdorff dimension of a
probability measure [I on C is defined as dim/J = inf{dim(X) : ”(X) = 1}. As m is

supported on J the question is to relate dim(J) and dim(m).

This problem has been solved by Zdunik in [Z] where she proved that if we asume
that the Julia set is not the all Riemann sphere we have dim(J) > dim(m) except for
the particular cases of critically finite maps with parabolic orbifolds. These consist -

up to Mobius conjugacies of the map f (z) = 2" and the Chebishev polynomials.

This problem can be solved in a more general setting of generalized polynomial-like
mappings. These will be our objects to study so we proceed with their definition: we
will be considering triples (f, V, U), where U is a topological disc and V is the union
of topological discs V1, . . . , Vk whose closures are contained in U. Also f,- drif f IV,- is a
regular or branched covering V,- —) U of degree d,- (so “regular” means that d.- = 1).

By (1 = (11 + + dk we denote the degree of the map f : V —) U. These dynamical

systems will be called generalized polynomial-like systems or GPL.

If k = 1, d _>_ 2, we come to the class of polynomial-like systems PL introduced in

DH and playin an important role in classification of polynomial dynamics.
g

Being GPL means to be quasiconformally equivalent to a polynomial:

fEGPL=>3thc(U):f=h-lopolyoh. (0.1)

This shows that many metric properties of K f 2f flux, f‘"(U) and J f = 0K;

are the same for a certain polynomial. This is just because metric properties such as
“zero area”, “absolute area zero”, “removability” or “uniform perfectness”, etc. are

qc—invariant .

Repeating the proof of Zdunik one finds that the dim(J) = dim(m) for a GPL if
either it is a PL with polynomials described by Zdunik or fm : V,- —+ U are linear
maps with the same derivative. The idea of the proof is to apply the Bowen-Ruelle—
Sinai thermodynamic formalism (see [Bo] , [Ru] , [PP] ) to conclude that either the

dim(m) < dim(J) or we have the homologous equation:

log|f’|5—logd=uof—u (0.2)

for some u E L2(J,m), where (I is the Hausdorff dimension of J. The analysis of

the homologous equation will lead to the above particular cases.

Here I f’ I‘5 is the Jacobian of the Hausdorff measure and logd is the Jacobian of
m. The term Jacobian is used for the derivative with respect to a measure. If a is
an arbitrary probability measure on J f then a general theorem ([Pa]), says that there
exists Jacobian J“ = Ju(f) on a set of full measure a. It means that there exists
Y C J, ”(J \ Y) = 0, and an integrable function J“ such that for every E C Y on
which f is l-to-I onto f(E) we have p(f(E)) 2 IE Judy. If Y = J we say that a has
Jacobian. The Hausdorff measure is not a probability measure but still we are using

this terminology.

4

0.3 Hausdorff measure versus harmonic measure

It is much more complicated to relate the harmonic measure and the Hausdorff mea-
sure and there is an extensive literature dealing with this problem. I would like to
mention first a result of general character, without dynamical context. This is a fa-
mous theorem of Makarov in [M1] showing that the Hausdorff dimension dim(w) of
the harmonic measure an on the boundary of a simply connected domain is always
equal to 1. So no matter how ”fat” is the boundary, the harmonic measure is always

hiding in a small l-dimensional subset.

In the same spirit, a result of Jones and Wolff shows in [JW] that dim(w) S 1 in
the general (not neccessarily simply connected ) case. Makarov’s approach was based
on the study of boundary distortion properties of conformal maps while Jones and

Wolff studied the behavior of Green’s function.

In the case of self similar sets the methods of ergodic theory can be relevant as it
was shown by Carleson in [Ca1],[Ca2] and Makarov in [M2]. This approach was used
by Przytycki in [P1] and by Przytycki, Urbanski and Zdunik in [PUZI] and [PUZZ]
for connected boundary, by Makarov and Volberg in [MV] ,and finally by Volberg in

[V01],[Vo2] for totally disconnected sets.

Here I would like to recall a problem of Volberg concerning the relation between
dim(w) and dim(J) for totally disconnected fractals. Consider a GPL (f, U, V) with

the assumption that the map flyi : V,- —-> U are conformal.

Then the Julia set will be totally disconnected and we are situated in the classical
case of hyperbolic dynamics . Volberg conjectured that in this situation we have
dim(w) < dim(J). According to this conjecture there are no exceptions: the harmonic

measure is always different from the Hausdorff measure.

The above inequality was actually proven by Volberg in two particular cases : in
the first case when we assume that the maps fm : V, —-) U are linear ( see [V01])
and the second case when we assume that certain symmetry conditions on (f, U, V).
The approach here is similar to the one in the work of Zdunik: by ”thermodynamic
formalism” we obtain that either we have the strict ineguality or the homologous

equation:

log |f’|“ — log Jw = u o f — u (0.3)

where 6 is the Hausdorff measure of J as before.

The analysis of this second homologous equation (0.3) leads to a contradiction

showing that only the first case is possible when we get the strict inequality.

I would like to call the attention to the fact, that a major technical difficulty arises
because in the homologous equation log d is replaced by log Jw. Therefore the analysis

of (0.3) is much more difficult.

In the work of Zdunik the measure of maximal entropy was so nice that there
was no restriction on the dynamics at all. The harmonic measure does not have a
good dynamical behavior generally and that’s why we need here the hyperbolicity

assumption.

Also we should mention here that in the above two problems the statement: ”Haus-
dorff dimension is strictly greater than the dimension of the maximal (harmonic)
measure” is equivalent to the weaker statement: ”the Hausdorff measure is singular
with respect to the maximal (harmonic) measure”. In principle when the Hausdorff
measure is involved the comparison of the dimensions is reduced to a problem of

absolute continuity of measures (compare [Le]).

The last pair of measures is the the harmonic measure and the measure of maximal

entropy. Their relationship is in fact the subject of my dissertation.

0.4 Measure of maximal entropy versus harmonic
measure

First we mention a result of Brolin in [Br] where it was established that the backward
orbits of a polynomial f are equidistributed with respect to w - the harmonic measure

on the boundary of the domain of attraction to 00 .

Almost twenty years later the ergodic theory of rational maps started with the
work of Lyubich , Mafié (see [Lyl] and [Mal]), and Mafi’e , Freire and Lopes in [FLM].
In the light of these works the result of Brolin can be interpreted as the fact that for

polynomials we have u) = m.

After a couple of years Lopes proved in [L0] that the converse is also true in the
sense that if we have a rational map f such that w = m on the boundary of an
attracting component of the Fatou set, then f is conjugated to a polynomial by a

Mobius transformation.

A simpler proof of this result was given in [MR].

0.5 The first question

t the starting point of this work we have decided to prove this result of Lopes under a
weaker assumption: w z m. In other words the following question arises immediately:
is that true for a GPL (f, U, V) that w z m implies that (f, U, V) is conformally

conjugated to a polynomial ?

This is a question of rigidity. We cannot compare now the dimensions of the two

measures but we can ask whether the absolute continuity of the two measures would

imply this strong information on dynamics.

The converse is obviously true: if our GPL is conformally conjugated to a poly-

nomial then we have the absolute continuity of the two measures by Brolin’s result.

This problem has been investigated by Lyubich and Volberg in [LV]. Under the
assumption of the hyperbolicity condition, the theorem of Lyubich and Volberg covers
two particular cases: when the map is PL and when we assume a symmetry condition

similar to the one in [V02].

In both situations we have that w z m implies that (f, U, V) is conformally
conjugated to a polynomial. This means that in a neighborhood of J (0.1) holds with

h being conformal.

As it was expected the homologous equation is connected to this problem looks

like:

logd—longzuof—u (0.4)

Because of the presence of the Jacobian of harmonic measure the same kind of

difficulty shows up as in the works [V01] and [V02].

Surprisingly the answer of this first question is not always positive , as an example

 

is given in [LV] where this property fails. Namely a GPL: (f, U, V) is constructed

where w = m but f is not conformally conjugated to a polynomaial.

0.6 The second question

Therefore we change the problem in the following way : first we call a GPL system

(f, V, U) maximal if m} = C0].

Next we call a GPL system f conformally maximal if it is conformally equivalent

to a maximal system, that is

f=h_logoh,

h : Uf —> U9 is a conformal isomorphism and (9, V9, U9) is a maximal system.

The basic problem in my dissertation is to prove that wf z m f implies that

(f, U, V) is conformally maximal.

We manage to prove this for the fairly general class of semihyperbolic GPL assum-

ing that the Julia set J is totally disconnected.

Here the semihyperbolicity of our GPL simply means that it is quasiconformally
conjugated to semihyperbolic polynomial. A polynomial is called semihyperbolic if all
the critical points which are in the Julia set are not recurrent and in addition we

assume that there are no parabolic periodic points.

I suspect that the result on conformal maximality is true generally, for any GPL
, without the semihyporbolicity assumption and whithout the assumption of J being

totally disconnected.

The main difficulty in this problem comes from the existence of critical points
in our Julia. Once we move out of the classical frame of hyperbolic dynamics the
Sinai-Bowen-Ruelle formalislm is no longer available. The other difficulty is that har-
monic measure,or equivalently Green’s function does not have apriori good dynamical

properties.

0.7 Outline of main results

In the first chapter, the class of separated semihyperbolic polynomials is introduced
and geometric properties of the Julia sets of such polynomials are studied. A subclass
of the separated semihyperbolic GPL consists of semihyperbolic GPL with totally

disconnected J.

In the second chapter we apply the results iof the first chapter in the above totally
disconnected case. Namely, we use these goemetric properties to conclude a good
boundary behaviour of Green’s function. This implies good dynamical properties for

harmonic measure and yields the solution of the problem on conformal maximality.

In what follows I will give a brief account of the results in each chapter.

In the first chapter metric properties of J are investigated. Since the metric
properties are preserved under q.c. maps and because of (0.1) we need to deal only
with polynomials instead of GPL. By results of Mafié in [Ma2] and Carleson, Jones
and Yoccoz in [CJ Y] the semihyperbolic polynomials have good distortion properties

implying a nice geometry of J.

Namely, a recent result of Carleson , Jones and Yoccoz says that a polynomial
f : C ——) C is semihyperbolic if and only if the domain of attraction to co , denoted
by Aoo( f) is a John domain. John domains are the objects what we need in our study.
They were thoroughly investigated in the simply connected case by many autours.

For their properties we can refer to [P02],[NV] and [CHM].

Here, we need a stronger - the so called uniformly John property. This roughly
means that John arcs are ”almost geodesics” in the internal metric. The main result

in the first chapter says that the uniformly John property of A00( f) is equivalent to

10

f being separated semihyperbolic.

To define this class of polynomials ,we denote by K; the filled Julia set and for
a: E K f let Kx be the connected component of K j containing at. Next we split the

set of critical points which are in K; into two parts :
01 = {col 6 Ky :wl is a critical point of f with Kw, = {w1}}

$22 = {0.22 E K, :w2 is a critical point of f with Kw, 7t {w2}}.
With this notation we have the following definition and theorem:

Definition 1 A semihyperbolic polynomial f : C —) C is called separated semi-

hyperbolic if there exists 6 > 0 such that:
dist(fn(w1), KW) > d for n E N,w1 6 ”had; 6 92.

A

Theorem 2 A semihyperbolic polynomial f : C —> C is separated semihyperbolic

if and only if Aoo(f) is a uniformly John domain. A

The proof uses the good distortion properties of the separated semihyperbolic

polynomials which imply a high degree of self similarity of the Julia set.

The uniformly John property will give a Carleson-type grid: a nested sequence of
John domains around each point of the Julia set. This is similar to the localization

theorem proven by Jones in [J01] for nontangentially accessible domains.

Also here in the first chapter we give examples showing that the following inclu-

sions are proper:

critically finite polyn. Q separated semihyperbolic polyn. Q semihyperbolic polyn.

11

The construction uses quasiconformal surgery and the fact that the filled Julia
set of a semihyperbolic polynomial has countably many nontrivial components. The

latter is proven by the technic of puzzle pieces of Branner and Hubbard (see [BH]).

Also our considerations show that for semihyperbolic polynomials the following
conjecture of Branner and Hubbard holds: the Julia set is totally disconnected if and

only if there are no periodic nontrivial components of the filled Julia set.

The second chapter starts with a necessary and sufficient condition for conformal

maximality.

Theorem 3 Let (f, V, U) be a GPL system. Two assertions are equivalent:

1) (f, V, U) is conformally maximal;

2) there exists a non-negative subharmonic function r on U, which is positive and

harmonic in U \ K}, vanishes on K; and satisfies

r(fz) = dr(z). A (Aut)

The proof is essentially a combination of ideas from [LyV],and [L0]. The direction

2) => 1) is due to Alexander Volberg.

Next we prove the result of conformal maximality for the case of PL or in other
words , for the case of connected Julia set. In this situation via the Riemann mapping
theorem our result will become a rigidity theorem of Shub and Sullivan in [SS]. This
illustrates the fact that our problem is a strict analog of the resuslt of Shub and
sullivan, but now we are living on totally disconnected sets. In our case harmonic

measure plays the same role as the Lebesgue measure in the work of Shub and Sullivan

([33])-

12

For the totally disconnected case the natural substitute of the Riemann mapping

: the universal covering map does not help.

The way to proceed is to study the boundary behavior of harmonic functions
in uniformly John domains. In addition to the localization property proven in the
previous chapter, we also use that the Julia set is uniformly perfect. This notion
was introduced by Pommerenke in [P02] meaning a certain thickness in the sense of
potential theory. Let us remind that the set E is called uniformly perfect with UP
constant a > 0 if

cap(B(x,r) O E) 2 a - cap(B(x,r)).

For Julia sets uniformly perfectness was proved independently by Hinkannen in [Hi],

Eremenko in [E] and Mafié with da Rocha in [MR].

Let Q be a uniformly John domain with uniformly perfect boundary and denote
by pg the internal metric in It. The next result is due to Volberg (see [BVI] and
[BV2]) and it will be essential for us. It is called the Boundary Harnack Principle

(BHP):

Theorem (Volberg) Let u,v be two positive , harmonic functions in II and
vanishing on U0 (1 50 where U0 is a topological disk. There exists 6 > 0 such that for

any topological disk U1 with 71 C U0 there is a constant K = Kymy, such that:

 

 

 

— 1] S K-[pg(x,y)]E forx,y E U109

A

This result is a generalization of a result of Jerison and Kenig in the case of

nontangentially accessible domains.

To come back to our original problem of, we start with a theorem proven by

13

Przytycki in [Pr3] and independently by Urbanski and Denker in [UD]. This is a

result of ergodic theory for any rational map, without assumptions of hyperbolicity.
The theorem says that if f : C —> C is a rational map and (,0 : J —> R is Hblder

continuous, then there is an equilibrium state ,u for go if the condition:

P(fM) > $11er (0-5)

holds ; where P( f, (,9) is the topological pressure of the function cp.

Quite naturally, our first result here says that if f is semihyperbolic the condition

(0.5) is no longer necessary; namely we have:

Theorem 4 Suppose that f : C —> C is semihyperbolic rational map and cp : J —>
R is Holder continuous on the Julia set J. Then there is a unique equilibrium state

p associated to the potential cp. A

This actually shows that the ergodic theory of semihyperbolic rational maps is

really the same as in the hyperbolic case and certainly this is true for semihyperbolic

GPL.

At this point we can apply Theorem 4 as soon as we check the Holder continuity

of (,0 = —log Jw.

Remind that we are in the totally disconnected case: it turns out that the internal
metric will be ”the same” as the euclidian metric. Then Volberg’s theorem gives the

Hblder continuity of (,0.

Now Theorem 4 gives that if (.0 z m we must have u = m and the homologous

equation (0.4) follows.

Finally using ideas from [V02] and [LyV] ,it is not very difficult to see that the

14

homologous equation (0.4) implies the result on conformal maximality:

Theorem 5 Let (f, U, V) be semihyperbolic GPL such that J is totally discon-

nected. Then m z w implies that (f, U, V) is conformally maximal. A

The proof is based on the application of Theorem 3. Staring with the homologous

equation (0.4) we are going to construct an automorphic function r.

Chapter 1

Geometric localization, uniformly
John property and separated
semihyperbolic dynamics

1 .0 Introduction

In this chapter we forget about our original problem and we are going to deal with
metric properties of Julia sets for polynomials. These properties will be the same for
GPL and we are going to use them in the second chapter where we investigate the

properties of harmonic measure.

Metric properties have their own interest: after all these are features that we can
”really see”. I would like to mention a property expressed by a procedure called

geometric localization.

Different types of geometric localization are extensively used in analysis. Local-
ization grasps fine properties of the boundary which allows to carry out estimates of
harmonic measure, Green’s function, etc. In the absence of the Riemann mapping

theorem localization may serve as its weak substitution.

The examples of such an approach can be found in Ancona [A1], [A2], J .-M. Wu

[W] for (mainly) Lipschitz domains and in Jones [J01], Jerison and Kenig [JK] for
15

16

non-tangentially accessible domains. Carleson’s work [Ca3] may be considered as a

source for this approach.

In this chapter we are going to deal with the localization property for John do-
mains. Our motivation is the following: let us denote by Aoo( f) the domain of at-

traction to 00 of a polynomial f. A recent result of L. Carleson, P.W. Jones and J .C.

Yoccoz ([CJY]) shows that Aoo( f) is a John domain if and only if f is semihyperbolic.

In the first section we prove the localization for simply connected John domains.

In Section 1.2 we give an example showing that the localization fails for arbitrary

John domains and we prove the localization at a fixed scale.

The third section is devoted to some geometric properties of the Julia set of a

semihyperbolic polynomial.

In the fourth section we introduce separated semihyperbolic dynamics and prove
that the localization of A00( f) is equivalent to the property that f is separated semi—
hyperbolic. For example localization works for critically finite f. Or in the case when

the Julia set is totally disconnected.

In Section 1.5 we show that localizability is equivalent to uniformity for John

domains and

Section 1.6 provides the example of semihyperbolic but not separated semihyper-

bolic polynomials.

In Section 1.7 we sow that a semihyperbolic Julia set is a boundedly finite- to-one
factor of the standard shiftspace. This dynamical property will be of use in the next

chapter when we study equilibrium states on J.

17

Commenting on Sections 1.1-1.3 let us mention that throughout them we play

with ideas virtually present in [CJY], [HR], [NV], [P01].

1.1 Localization of simply connected John
domains

Let Q Q (C be a John domain. This means that there is a point x0 6 9 (called

center) such that any x E It can be connected to 1:0 by an arc 7 Q 9 such that:
dist (6,30) 2 c dist (fix) for {E 7,

and some c > 0 independent of x.

The arc 7 is called John arc and the best constant c > 0 is the John constant of
the domain It. A simply connected John domain is called a John disk. Let us recall

that the internal metric p for a domain It is defined by:

p(x, y) = inf,{diam 7 : 7\{x, y} C Q is a curve connecting x and y}, for x, y E D.
In the case when It is simply connected p can be extended so that (9*, p) becomes a
complete metric space where (1” denotes the union of the interior points and prime
ends. In what follows we denote by Bp(Q, r) the points in It which r-close to Q in

the metric p.

1.1.1 Definition. Let Q be a John domain and Q E 39, r > 0. A finite collection
of domains {flghfllzm is called a John prelocalization at the point Q on the scale

r if:
(1) DEV) C It is a John domain with constant c for 6 == {1, . . . , N}

(2) U( ”(QU) 3 Bp(QaT)

(3) diam (28(7') 3 M-r, €={1,...,N}.

18

1.1.2 Definition. A prelocalization is called localization if in addition to (1),

(2), (3) we have
(4) our) 00%) = (a, i at]; Q e an, o < r < r0.

1.1.3 Definition. A John disk admits localization for all scales r < r0 if there

is a localization at any point Q E 30 for all scales r < r0, moreover the constants

N, M,c do not depend on Q and r.
The main goal of this section is to prove Theorem 1.1.9 but first we prove

1.1.4 Theorem. A John disk admits prelocalization. A

First we give a series of preparatory results. They are known and we include them

for the convenience of the reader.

1.1.5 Lemma. Let f : ID —> It be a conformal map onto a John disk 9. If
Q’ (_2 ID is a John disk and dist (f(0),8f2) 2 c diam (Q) then f(fl’) is a John disk

with constant depending only on Q, Q’ and c. A

Proof. For 2 6 ID, 2 = re‘it let us introduce
8(2) 2 B(re“) = {pew : r S p S 1, [0 — t] S 7r(1— r)}.
The proof will be based on the following characterization of John disks (see [P01],

p.97):

Theorem A. Let C map I) conformally onto G such that dist(C(0),3G) 2 c1

diam(G). Then the following conditions are equivalent:

(i) G is a John disk with constant c,

 

19

(ii) there exists a, 0 < a S 1 such that,

1-|€|
1“IZI

 

IC’(€)ISM1|C’(z)l-( ) forseB<z),zeD

(iii) diam C(B(z)) S M; dist (C(z),80) for z E G, where the constants M1,M2,a

depend only on each other, the John constant c and the constant c1. A

To start the proof of the lemma we consider the Riemann map g : ID —> 9’ such
that dist (g(0),80’) Z c’l diam (Q’) where c’1 depends only on c’ - the John constant
of (2’. Consider the map h = f o g, h :ID —> f(fl’). According to Theorem A we need
to estimate

h'(é) = |f’(g(€))l ° lg'(€)l for 66 3(2), z 6 D-

By Theorem A:

diam g(B(Z)) S M2 dist (g(z),BID). (1.1.1)
To estimate |f’(g(§))| we distinguish two cases: when diam g(B(z)) 2 g and diam g(B(z)) <
%. In the first case define 2 = 0 and in the second case define :7: = g(z) - (1 —

2 diam g(B(z))). In the first case by (1.1.1) we have dist (§,g(z)) = dist (0,g(z)) S

1 — Ml? Then by the distortion theorem we have

|f'(5)| = lf’(0)| S Malf'(g(2))|- (1-1-2)

In the second case dist (§,g(z)) = 2 diam g(B(Z)) S M2 dist (g(z),81D) and again
by the distortion theorem:

lf’(5)| S M3]f'(9(2))|- (1-1-3)
It is no loss of generality to assume that Q' Q ID is included in a half disk. Then if

we consider 3(3) Q ID we observe that in both cases g(B(z)) Q 8(2). Now we apply
Theorem A for the function f, and for g(§) E 8(5):

If’(g(€))l s Mum»- (1312954) _ .

- IZI

20

Because 1— lil S M5(1— |g(z)|) by the way E was defined we get:

1— lg(€)l
1 - |9(Z)|) '

By the Koebe distortion theorem (considering two cases: either dist (g(z),BII) S or

2 6 - (1 - lg(2)|)
l—ltl)
(1-...) -

1—IZI

was» s Melf’(§)l- (

9;“).

If’(g(€))| S M7|f’(5)|- ,
9 (Z)

 

 

If we apply Theorem A for g we obtain:

 

lf’(g(6))llg’(€)| s Maura- (

and by (1.1.2) or (1.1.3) we get:

 

I _ |€l)a(a -l)

Ih’(€)l s M9Ih’(z)|- (1 _ ,2,

which is the estimate we need to conclude that
h(lD) = f(fl’) is a John domain.

Notice that the constant M9 in the final estimate depends only on the constants of
f and g and hence the John constant of f (9’ ) depends only on c and constants of Q

and 9’ respectively. I

Following [NV] a John disc (I has locally connected boundary. If f : ID —> It is
the Riemann map we can consider the continuous extension f : ID —> IT. The set of
prime ends 0*9 is extremely simple. The impression of each prime end contains one

point and moreover the prime ends are just the accessible points.
The following lemma tells more about the boundary of (2.

1.1.6 Lemma. Let (I be a John disk with constant c > 0, f :ID —-> It the Riemann

map extended continuously to f : ID —~) D. Then for Q E 8Q,

#(f—1(Q)) S K where K = K(c). A

21

Proof. Let Q E 80, z,- E f‘1(Q),i = 1,...,K. Consider the geodesics 7,- =
f([0,z,]). Let us use the convention 7K+1 = 71. Then 7,- U 7,11 are closed Jordan
curves for i = 1,...,K. Because 7,- is not homotopic to 7,-4.1, there exists y.- 6
int(7,- U7,+1) 0 9°. Since 9“ is connected there is a continuum E,- C QC flint(7,- U7,+1)

connecting y,- and Q.

Let r = min,- Iy, — Q]. Clearly E,- intersects BB(Q,r) in certain points which we

denote again by y;. Similarly let us denote by y, the intersection 7, 0 BB (Q, r).

Consequently we have obtained 2K points yl, y1, y2, y2, . . . ,yK, yK situated in this
order on 88(Q, r).

By the fact that the geodesics 7, are John arcs (see [GHM], [NV]) we obtain that
B,- = B(y’hc-r) C (I.

Finally if K > [27”] + 1 then we necessarily have that y,- E B: for some i which is

in contradiction to the fact that B,- C II. I

Before the next lemma for Q E 09, r > 0 introduce the notation Ora for the
union the components of 0 fl B(Q, r) which have Q at their boundary. Furthermore

notice the inclusions

Qr‘Q C Bp(Q,2T) C anQ.
Before our next result let us introduce some notations.

Given (1 > 0 let {z,}i:fi = f‘1(Q) for Q E 30, 5.; = r,- - 2,, 0 < r,- < 1 where r,-
is chosen such that dist ”(50,351) = d, i = 1,K. Let B(§,) as in Lemma 1.1.4 and
T,- = 6f(B(§,-)) \ 89. With these notations we have:

1.1.7 Lemma. There exists c1 > 0, c1 = c1(f2) such that:

“cl-dc C Uf(B(5i))- A

22

Before the proof let us state the following

1.1.8 Corollary. There exists c1 > 0, c1 = c1(Q) such that:

B.(Q,c. -d) c Uf(B(2.-)). A

Proof of Lemma 1.1.7. It would be enough to show that T,- = 0f(8(§,~)) \ 00
is far from Q. Unfortunately this is not the case but we will construct a subdomain
D.- g B(§,) such that r.- = 8f(D,-) \ an is far from Q. To do this denote by 11,12 the
two halves of 38(2,) 0 BID. Because diam f(Ij) 2 c2 - d (see [P01], Prop. 4.19) for
j = 1,2; there exist Q1 6 f(ll), Q2 6 f(Ig) such that:

Cg’d
IQj—QIZ

 

, j: 1,2. (1.1.4)

Denote by Pj E f ‘1 ( Q j) the closest preimage to 2,, j = 1, 2, and consider the ray from
Pj to the boundary of B(Ei). In this way we constructed a subdomain D,- Q 8(E,-)
bounded by the two ray segments RI, R2 and arcs C1,C'2, where Cl C 68(2)) \ 81D

and C2 Q alD. Take a point x 6 C1 then:
dist (f(x),Q) Z dist (f(x),6SI) 2 c1 - (1.
Suppose x 6 RI, then:

dist (f(x),Q) Z dist (Q1,Q)- di3t(f($).Q1)- (1-1-5)

On the other hand f(Rl) is a John arc ([GHM], [NV]) and therefore:

dist (f(x),Q1)S 3 dist (f(x),Q)-

C

This estimate and (1.1.5) imply:

dist (f(x),Q) 2 C3 ' diSt (Q1, Q)

23

and by (1.1.4) we get:

dist (f(x),Q) 2 c1 -d.

The estimate is the same for x E 8.2 and we are done. I

We are now in position to give the:

Proof of Theorem 1.1.4 The proof is a combination of the previous results.
Let f : ID —> 9 be the Riemann mapping. For Q E (99 consider {2:}:in = f'l(Q)
as in Lemma 1.1.5. If Cl is the constant from Lemma 1.1.6 put d = g and define:

'Q(r) = f(8(2,)), i = 1,_K. By Lemma 1.1.4 Star) are John domains with constant

c depending only on Q.

It is clear that {08(r)}£=1—,7‘7 satisfies (1) and (3) with N = K, M = i. By

Corollary 1.1.8 property (2) holds also. I

The following is an easy consequence of the previous results:

1.1.9 Theorem. A John disk admits localization. A

Proof. The idea is to enclose two subdomains which intersect each other by a
slightly bigger domain. So let us start with Star), Star) such that Q'QU) fl {28(r) #
(0. Then with the notations of Lemma 1.1.6 we have 8(2,-) (1 8(2,) 74 (0. Suppose
diam 8(2,) 2 diam 8(2,) and enlarge 8(2,-) to obtain a new domain 8(2) with
8(2) _12 8(2,-)U8(2j) and diam 8(2) S 4 diam 8(2,-). It is clear that dist (f(2), Q) S
M,r.

Define a new collection {98(7‘)}(=1,N_1 where we replace (22,0) and 96(r) by
f (8(2)) The new collection has all the properties (1), (2), (3) but now the constant

M needs to be replaced by M1 - M. If the domains are disjoint in this new collection

24

we stop, if not we replace two by one as above. After at most N number of steps we

either stop when we get disjoint domains or we obtain just one domain.

The consequence of this process is a localization since the domains will be disjoint

and we changed the constant in (3) to at most M - M1N.I

In the next section we deal with arbitrary John domains, not just John disks.

1.2 Localization for arbitrary John domains at a
fixed scale

At the end of this section we say a few words on the comparison of our localization
and the one in [CJY]. First we present an example which shows that arbitrary John

domains do not admit localization.

1.2.1 Example. Let (20 = {x + iy : x 6 [—1,1], y 6 [0,2]} and consider (I =
(to \ UCn where CT, = {iy : y}, S y S yfi}. We choose gig: such that y}: > y}, >
Elli-+1 > yk+1 and y: "' 971; Z 2—11, 9111 — 97214-1 : 2_n2-

It is easy to see that It does not admit a localization at the point 0 E 80 because

the John constant will tend to 00 as the scale tends to zero. A
However, an arbitrary John domain admits localization if the scale 5 > 0 is fixed.

Namely we have the following:

2.2.2 Theorem. Let 8 > 0 be fixed and Q be an arbitrary John domain. Then II

admits localization at the scale 5 > 0. A

Proof. Since we have a localization in the simply connected case it is natural to

25

add boundary pieces to 80 to obtain a simply connected John domain fl. Using a
result of Jones (see [J02], Theorem 2) we can do this without changing too much the

John constant of II.

Applying Theorem 1.1.9 to D we obtain a collection (QE(€)}¢=1,N of simply con-

nected John domains with properties:

 

(1) diam (222(5) S M0 - e, t =1,N

(2) UQHS) 2 Bp(Q~;5)

(3) 022(8) 0 (28(5) 2 (l) for i 76 j.

Next we are going to eliminate the boundary pieces we have added for the construction
of fl. Let us introduce some notations to indicate the presence of these boundary

pieces. Let c‘ be a center of 022(5). For a pair (i, j) of indices introduce:

F(,,J-),Q,M = {7 Q (I : 7 connects 6,0" where cf, of are centers offlae), 08(6)

and diam 7 S 2M - 6}

Furthermore we define:

0 If P(g,j),Q,M = 0
C(i,j)(QaM) = sup inf dist (5,80),

WEF(‘»J)Q;M ”67

and let C(Q, M) = max(,-,j) C(i,j)(Q, M).

Let us put M = MD with Mo from (1). The indicator c(Q,Mo) shows whether
these might be boundary pieces we need to eliminate. In particular if C(Q, Mo) = 0

we have:

(8022(5) n 095(5)) \ on = u

26

for any pair (i,j).

Let F = {Q E 39 : c(Q,Mo) > 0}. The proof of the theorem is based on the

following simple fact:

1.2.3 Lemma. There is a = (1(9) > 0 such that c(Q,2Mo) > a for any

QEF. A

Proof. To see this let Q1, 6 F be such that c(Q,,, 2M0) —) 0. Let 9bn(e), 9i "(8)
be such that c,-j(Q,,, MO) > 0 and CL, c3, be the corresponding centers. We can assume '
that Cf, —> c‘, c}, —+ cf. Then there is no such that dist (ciomi) < 924‘, dist (c’no,c,’,) <
9124‘- for n 2 no. Here cf, of may not be the centers. We do not care.

Let 8 = C(Qno, M0) and consider the curve 7,, = [cf,0,c:,] U 7no U [01,20,031] where
7no is connecting cio to cf") with diam 7,,o Q 2Moe, and dist (6, 39) 2 g for E E 7%.
Then 77, connects cf, to cl], diam 7,, S 4Moe and dist (E, 39) 2 8 where 8 =

min(€-, 92E). Therefore 7n 6 P(i.j),Qn,2Mo and c(Q,,, 2M0) 2 8 which is in contradiction

to c(Q,,,2M0) -> 0. I

Proof of Theorem 1.2.2. We can now proceed with the elimination of the
extra boundary pieces. This algorithm will be used in later sections where we do the

localization at any scale.

If C(Q, Mo) = 0 there is nothing to do for this particular Q E 39. If C(Q, Mo) > 0
take a pair (i,j) such that c(,-j)(Q, Mo) > 0 and hence c(,-,J-)(Q,2Mo) Z a. Let 7;,- be

a curve connecting ci and of in 9 such that diam 7,,- < 4M0 - e and dist (5,39) 2 %

for 5 E 7.7-. Let ng be a neighborhood of 7,, of thickness %. Then Hij Q 9 and we

can define:

931(5) = intan‘ga) u H.,- u 62(5)) \ an).

27

It is clear that 9g (5) is a John domain with the correct constant and by this step we
have eliminated the common boundary (395(5) 0 396(5)) \ 39. Consider the new
system of local domains where 95(5) and 9i) (5) are replaced by 9'0] (5) and denote
this again by {95(5)}5. The new system has all the properties of the old one, the
only difference is that M0 in (1) is replaced by 5M0 and we reduced the number of

domains by one.

If c(Q, 5M0) = 0 for the new system then we stop. If c(Q,5Mo) > 0 we perform
the above construction again reducing the number of domains. We repeat everything
at most N times and as a result either we will stop when c(Q, 5kMo) = 0 for some k,
0 S k S N or we will have just one domain left. In both cases we obtain a collection

of local domains {922(5)}; which is a localization at scale 5. I

1.2.4 Remark. In the proof of Lemma 1.2.3 we had implicitly that a = a(9,5)
and therefore the John constant also depends on 5. This in fact happens for the

domain in Example 1.2.1. A

1.2.5 Remark. It is interesting to compare localization in Theorem 1.1.9, Theo-
rem 1.2.1 and the one in (5.1) - (5.4) of [CJY]. Basically (5.1), (5.2), (5.4) are obtained
in Theorem 1.1.9. But not (5.3). Localization of Theorem 1.4.9 below is much finer
than (5.1), (5.2), (5.3), but again does not touch (5.4). However, one can prove that
localizations of Theorems 1.1.9, 1.2.1, 1.4.9 satisfy (5.4) of [CJY]. But here in this
chapter we are not using harmonic measure or Green’s function, concentrating only

on geometry.

In the next section we are going to present some geometric properties of the Julia
set J of a semihyperbolic polynomial. A polynomial is called semihyperbolic if it has

no parabolic periodic points and for any critical point w E J we have dist(w, o(w)) > 0

28

where 0(w) denotes the forward orbit of w.

Let Aoo( f) be the domain of attraction to 00 for f. A recent result of Carleson,

Jones and Yoccoz states that Aoo( f) is a John domain if and only if f is semihyper-

bolic.

Our next purpose is to show that in some of the cases Aoo( f) admits localization

and to characterize all such cases.

1.3 Geometric properties of the Julia set

Let f : U —> V be a polynomial where U C V are topological disks. Then J Q U

and we always assume

(*) J = fl f‘"(U)-

n=0

Condition (*) means that the Julia set of the polynomial does not split the plane.

In what follows we are going to describe briefly a method used by Branner and

Hubbard in studying the structure of cubic polynomials (see [BH]).

Let 71,72 analytic Jordan curves such that U Q U1 Q U2 Q V where U1 =
int(71), U2 = int(72). Without loss of generality we assume that if L.) is a critical

point for f then either a) E J or w E V \ U2.

For x E J denote by Cw the connected component of J containing x. Let Pn(x)
be the component of f‘"(U2) such that C1. Q Pn(x). It is clear that f : Pn(x) —>

Pn_1( f (x)) is a branched or regular covering and we have the chain:
0x Q Pn($) Q Q P1($) Q U2-

By (*) it is clear that C1. = 2:0 Pn(x).

29

This proves by the way that for 5 > 0 there is an analytic curve 7 = 7,, disjoint
with J and surrounding 0,, such that diam 7 S diam Cx + 5. In fact for given 5 > 0

we put 7 2' 3PN(x), and take N to be large enough.

Let A0 2 U2 \ U1 and An(x) be the annulus surrounding x, An(x) = Pn(x) fl
f‘”(Ao). We are going to call An(x) critical if a critical point a: is surrounded
by An(x). In this situation we have An(x) = An(w). Furthermore f : An(x) ——)

An_1(f(x)) and we have the chain
An($) —* An—1(f($)l —> —+ ANN-1(3)) —’ A0-

The map f : An_,(fi(x)) —> An_,_1(fi+l(x)) is univalent if An_,(fi(x)) is not critical.
If this happens we have that mod An_,(f‘(x)) 2 mod An_,'_1(f"+l(x)). If this is the

case for any i, 1 S i S n we get mod An(x) 2 mod A0.

In the case when An-,(fi(x)) is critical the map

f = An—z'(f'i(€17)) —+ An—i-1(fi+l($))

is a regular covering of a certain degree d,- and we have mod An_,(f‘(x)) = imod An_,-1(fi+l(x)).

For example if there is just one critical point we of multiplicity 1 we have that

mod Ans-(ran = 3 mod Ans—Arron

in the case when An_,(f‘(x)) = A,,_,-(w0). If 1: denotes the number of critical annuli

among An_,(f‘(x)) then mod An(x) = gymod A0.

The following result is very useful in deciding whether 0,, = {x} for a certain x

(see [BH]).

1.3.1 Lemma. Suppose A is an open bounded annulus and An is an infinite set

of disjoint open annuli An Q A each one winding around the bounded component of

30

C \ A. If Zn mod An = 00 then the bounded component ofC \ A equals one single

point. A

To apply this lemma notice that since f‘1(U2) C U1 we have that An(x) Q
int An_1(x) where by int An_1(x) we denoted the inner component of C \ An_1(x).

Therefore {An(x)},, is a sequence of disjoint, nested annuli surrounding C1,.

If there is no critical point in J then mod An(x) 2 mod A0 and Zn mod An(x) =
00. By Lemma 1.3.1 we obtain C: = {x}. This shows that the hyperbolic Julia set

which is disconnected is always a Cantor set. Another application of Lemma 1.3.1 is:

3.2 Lemma. Let f be semihyperbolic and Cw = {w} for any critical point w.
Then Cr ={x} forx E J. A

Proof. The idea is to show that mod(An(x)) 2 c and then apply Lemma 1.3.1.

Let n E N and i(n) be the number of indices i such that An_,(f‘(x)) = An_,-(w)
for some critical point w. We claim that there exists D E N independent of n such
that i(n) S D for any n E N. If this it true then we can decrease the modulus at

most D times and hence mod(An(x)) 2 c.

Suppose our claim is not true. Then there is a critical point too such that if i(n)
denotes the number of indices i for which An_,-(f‘(x)) = An_,-(wo) we have that i —) 00

as n —> 00. Let i1 < i2 < < i,-(,,) be the indices such that

An—iJ(fiJ(-T)) : Ari—iJ (“20), tj E {ila - ‘ ° aii(n)}-

Because An_,,(f‘1(x)) = An_,-,(wo) we get that An-,,(fi2(x)) = An_,2(f‘2“1(wo)).
On the other hand A,,_,-2 (f‘2 (x)) = A,,_,-2 (wo). Consequently f‘2"1(w0) E P,,_,-2 (wo).
Denote i2 — i1 = jn. Then fj"(wo) E P _,2(w0). It is clear that n — i2 —> 00

as n —-) 00 and Pn_,-2(w0) —> Cwo. But Cwo = {we}, and consequently there is a

31

sequence of iterates {f3}: (wo)},, such that fj"(wo) —> we which is a contradiction to

semihyperbolicity. I

To deal with the critical points to for which Cw at {on} we need the following

characterization of semihyperbolicity due to Carleson, Jones and Yoccoz (see [CJY]).

First we introduce some notations: for 2 E C denote by 8,,(x,5) the connected
component of f ""(8(2, 5)) containing x. By the maximum principle 8,,(x, 5) is simply
connected and f" : 8,,(x, 5) —+ 8(2, 5) defines a branched or regular covering and we

denote by dg(8,,(x,5)) its degree. With this notations we have:

Theorem B. The following are equivalent:

(A) f is semihyperbolic

(B) there exists 5 > 0, c > 0, 0 < 0 < 1 and D < 00 such that for all x 6 J and

nEN

/\
b

Clg(B,,($,€))

diam 8n(x,5) S c-O". A

Using this result we can prove:

1.3.3 Lemma. Let f be semihyperbolic and x E J such that C1, 75 {x}. Then

{Cf"(r)}k is either periodic or preperiodic. A

Proof. Let x E J such that C1,. 75 {x}. First we are going to show that there
exists 6 > 0 such that diam Cjkm Z 6 for k E N. For if not, then there exists a
subsequence k,- such that limHoo diam C fie-(x) = 0. Then there is an index i0 such

that diam wax) S g- for i 2 i0. This implies that Cf“*‘(x) Q 8(f’“(x),5) for i 2 i0.

32

By Theorem B diam 8k,(x,5) S c- 0’“ and hence lim,_,oo diam 8k..(x,5) = 0. On the

other hand 8k,(x,5) 2 C1, which is a contradiction.
Suppose next that {C [kWh is an infinite sequence of distinct components.

Take a subsequence k,- such that Cw“) are all different and choose a further
subsequence kg} such that f k'1 (x) —> x0. In this way we obtain a point x0 such that
there is a sequence of distinct components of J of fixed size 6 accumulating at x0. On
the other hand U2 \ J is a John domain (see [CJY]) and this leads to a contradiction.

A consequence of this lemma is:

1.3.4 Corollary. Let 0,, 74 {x} be a component of a semihyperbolic Julia set J.

Then Cx is equal to or is a preimage of a periodic critical component Cw. A

Proof. Let us consider the annuli An(x) as in Lemma 1.3.2. From the proof
of Lemma 1.3.2 we see that the number of critical annuli i(n) tends to infinity as
n —) oo, otherwise we would have 0,, = {x}. Then there exists a critical point
we such that An_,-,‘(f""(x)) = An_,-k(w0) for k E {1,..,i(n)}. Among {Cf"(x)}n20
there are p distinct components and let X1,...,Xp be them. Among {C fnwo }n20
there are q distinct components and let C1 = Cwo, . . . , 0,, = C fq-le be them. Among
f"1 (x), . . . ,f‘P+1(x) there are two points, say f'i"1(x),f‘."2(x),i;cl < ikz, belonging to the
same Xpw, 1 S p(n) S p. Notice that ik, and ik, depend on n. Then An_,kl(f‘*1x)
and An_,k2 ( f‘*2 x) encircle X1201): Let us denote the first of these annuli by A? and
the second by A3; the corresponding pieces are called P,” and 82". Then Xp(,,) Q
P," Q P2". Now let N be an infinite set of indices n for which p(n) 2 pa, for a certain

Po, 1 S p0 S p, which clearly exists.

 

33

We can write the chain of equalities
A3 = Ame, U“? at) = fi*2_i*‘(An_i.,(fi*1$))

= f “2"“ (An—n, (wo)) = A.-.., (f ‘*2“"1wo)-

This means that fik2 "'"1we E Pf. But A" = A,,_,-k2 (we) and so also we E 85‘. Thus

both Cw, and Cf,2 _.,,l we lie in 82". Because it — ik, 2 n — ip+1 ——) 00 we can write

XPO : 0 P2" 2 CW0
nEN

It rests to prove that Cwo is periodic. If for a certain it we would have f‘*2 ""1 we 6 Cwo
we would be done. Otherwise for infinitely many n E N f {,2 ""Iwe E qu for a certain
qe, 2 S qe S q. Thus qu lies in P," for those n and so que—wa = 0,0 = flue” P; =

Cwo and we are done. I

1.3.5 Remark. At this point we can see that a conjecture of Branner and Hub-
bard is true for semihyperbolic polynomials. Namely we can conclude that J is a

Cantor set if and only if there are no periodic critical components. A

1.4 Localization of A00( f )

Theorem 1.4.9 is the main result of this section. But before proving it we are
going to give independent proofs of its particular cases. This seems to be illustrative.
In what follows we are going to use the distortion properties of d-valent functions.
First let us introduce some notations. Given a topological disk W and a closed set

F Q W we say that F is a—admissible for a > 0 if
diam F 2 a diam W.

For a given a > 0 we say that a domain W is a—thick at a point 2 E W if

dist (2,3W) Z 01- diam W.

34

We will use the following lemma which is in the spirit of Proposition 2.1 of [HR].

Lemma C. Let (W1, V1), (We, Ve) be two pairs of topological disks V, C W}, i =
0,1 and0 < 8 < mod(We\Ve). Fix a > 0, d E N and let f : W, ——+ We be a branched

covering of degree (1 such that V1 is a component off-104,). Then the following holds:

a or an two connected a-admissible sets F’, F” in V1:
31

diam (F') diam f(F’)
diam (F”) N diam f(F”)’

 

 

(b) the same holds for F’, F” such that f(F'), f(F”) are a-admissible and connected,

(c) for a point xe 6 V1 and a closed set F Q V1 we have dist (xe, F) ~ diam (V1)

ifdist (memo) ~ diam (vo)

(d) for a point xe 6 V1 and a closed set Fe C Ve we have dist (f(xe), Fe) ~ diam Ve

if dist (xe,f‘1(Fe) n V1) ~ diam V1. A

1.4.1 Remark. A consequence is that V1 is a-thick at a point xe 6 V1 if and only
if V}; is a’—thick at f(xe).

There are a number of particular cases when we can achieve a localization. The

first case is:

1.4.2 Theorem. Let f be semihyperbolic and assume that Cw = {w} for any

critical point w E J. Then Aoo(f) admits localization. A

Proof. By Lemma 1.3.2 J is a Cantor set in this case. Moreover the annuli
An_k( f k(x)) can be critical only for a finite number of indices k. Let 8,,(x) be the com-

ponent of f‘"(U1) containing x. The degree of the map f” : Pn(x) —> U; is bounded

35

by D, where D E N is independent on n and x. It is easy to see that Pn(x) is a-thick
at any point 2 E 313,,(x). Therefore dist (3Pn(x),3P,,(x)) ~ diam Pn(x). Consider
an analytic Jordan curve 7,, in An(x) such that dist (7n,3An(x)) ~ diam Pn(x) and
let 9,,(x) be the domain containing x with 39,,(x) = 7”. It is clear that 9,,(x)\J
is a John domain with the right constant. From Lemma C it follows also that
diam Pn(x) ~ diam Pn+1(x) and hence diam 9,,(x) ~ diam 9,,+I(x). Now for
given x E J, r > 0 put 9,,(r) = 9,,(x)\J where n is the chosen to be the largest

integer for which diam 9,,(x) 2 r. I

The next case is when CW 75 {w} for all the critical points w E J. In this situation
we are going to add extra boundary pieces to obtain a simply connected domain 9
as in Section 1.2. We consider the collection {95(7‘)}g=1,}v of simply connected John

domains and introduce:

F(i.j).Q.M.r = {7 : 7 connects 3,0”, where 3,0" are centers of 95(r),

9i r such that diam 7 S 2M - r
Q

As in Section 1.2 we introduce the control:

0 if F(i.j).Q.M.r = 0
, d. t ,J
76F(i,j)Q,M,r {6'7 7‘

c(,-,,)(Q, 7", M) =

and put c(Q,r, M) = maxed) c(,,,-)(Q,r, M).

Let F(r, M) = {Q E J : c(Q,r, M) > 0} and we have the following:

1.4.3 Lemma. Let f be semihyperbolic and assume that 0,, 75 {w} for any critical

point w. Then there exist M’ > 0, a > 0 independent on and r such that

c(Q,r,M’) >0: for any Q6 F(r,M). A

36

1.4.4 Theorem. Let f be semihyperbolic and assume that C,,, 56 {w} for any

critical point w E J. Then A0,,(f) admits localization. A

Proof. Lemma 1.4.3 is a key step and the localization now follows by the algo-

rithm described in Theorem 1.2.2. I

Proof of Lemma 1.4.3. Suppose the statement is false. Then there is a
sequence {Qn},,, {r,,},, of points and scales such that Qn E F(rn, M) and
c(Q,,,r,,,n) —> 0 as n —> 00. Let clue}: be centers of 910n(r,,),922n(r,,) and 7,, be
a curve connecting cincf, with diam 7,, S 2M - r,,. Let 5 > 0 be small and M, >
M2 > 2M3 large constants. Denote by W}, the component of f’k"(8(f’°"(Q,,), M,-5))
which contains Q,, and kn is the largest integer such that diam (W3) 2 K - M . r,,.
By the semihyperbolicity of f we can choose 6 so small that the map f"" : W: -->
8(f’°"(Q,,), M1 - 5) has degree S D where D is independent on Q". We can apply
Lemma C and conclude that W: is a’-thick at Q,,. Now choosing K big we conclude

that 7,, Q W3.

Without loss of generality we can assume that f k"(Q,,) —> P and let now W,',P, i =
1,2,3, be the component of f‘k"(8(P,2M,-6)) containing Q,,. Let us denote b}, =
f k" (c,l,), b3, = f k" (cg) and without loss of generality again we say that b1 = lim bi, b2 =
limbfi. If n is large enough, say it 2 ne, W,', Q Wile, i = 1,2,3, and by Lemma C
dist (P,b},) ~ dist (P,b3,) ~ 6, dist (b3,,J) ~ 6 ~ dist (b§,J). Let 1“,, = fem)
connecting b}, to bi. As 7,, Q W: Q Wip we have I), Q 8(P,2M36) and hence
diam I‘n S 4M3 - 6. By the algorithm of Lemma 1.2.3 there exists Pn connecting b},
and b: (n 2 no) such that diam I), S 8M3 - 6 and dist (§,J) 2 a” for E E P".

Let Ln = I‘,, U P". Let us recall that by Lemma 1.3.3 there exists 6’ > 0 such that

diam (ka(w)) > 6’ for any k E N and any critical point w. Choosing 6 small with

37

respect to 6’ we obtain that the index of L,, with respect of any critical value is equal
to 0. This implies that there is an f "Mlifting 7,, of I”, which has endpoints c}, and CE].
On the other hand by Lemma C diam 7,, S 2M’ - r,, and dist (5, J) 2 oz - r,,, 6 E 7,,,

which contradicts the fact that c(Q,,,r,,, n) —> 0. I

1.4.5 Corollary. Let f be semihyperbolic with only one critical point on J. Then

Aoo(f) admits a localization. A

1.4.6 Corollary. [ff is a cubic semihyperbolic polynomial then A0,,(f) admits a

localization. A

In fact, there are three possibilities: 1) both critical points escape to infinity
and then hyperbolicity implies localization; 2) both critical points are on J, so J
is connected and localization follows from [CJY] and Theorem 1.1.9; 3) one critical

point is on J, another escapes, then Corollary 1.4.5 implies localization.

1.4.7 Remark. In Section 1.5 we introduce the notion of uniformly John domain
and prove that uniformly John domains are exactly localizable John domains. After
that we can reformulate the results above as follows: under certain condition (e.g.
only one critical point on J) Aoo(f) is a John domain if and only if it is a uniformly

John domain. A

The following result is actually due to Volberg but we include it here as an inter-

esting particular case.

1.4.8 Theorem. Let f be critically finite. Then Aoo(f) admits a localization. A

Proof. We use the notations c(Q,r, M), f(r, M) of Lemma 1.4.3. And we are

 

38

going to prove as before that
3M’, 01 > 0 : c(Q,r, M’) > a for any Q E f(r, M)

Let us remind that this statement implies Theorem 1.4.8 by the algorithm used in

Theorem 1.2.2.

First we repeat wordly the proof of Lemma 1.4.3. If the statement is false then
there is a sequence {Q,,}, {r,,}, Q,, E f(rn, M) and c(Q,,,r,,,n) —> 0 as n —> 00. We
consider c1,,c,2,,9bn(r,,),9gn(r,,) and 7,, connecting C1,,c1", with diam 7,, S 2M,, - r,,.

We construct k,, and W,1, i = 1,2,3, with the help of M, > Me > 2M3 and a small 6

which we are going to choose from the condition that

in any disk of radius 2M1 - 5 centered at J there is at most one critical

value of f.

We introduce W,’,’P, i = 1,2,3, exactly as before and b1 = Iimbi, b2 = lim b3, where
b1, 2 f’c"(c,1,), b3, 2 f""(c3,). By Lemma C dist (b1,J) ~ 5 ~ dist (bfi,J). Let
F,, = f"“ (7,,), it connects b}, to bi. As 7,, Q W3 Q WE’P we have F,, Q 8(P,2M35)
and hence diam I‘,, S 4M35. Modify I‘,, to F1, = [b1, b1,]UI‘,,U[b?,, b2], curves connecting

b1 and h”. Then diam F1, S 5M36, n 2 ne.

Let y be the (possible) critical value of f in 8 (P, 2M, -6). For the sake of simplicity
we assume in what follows that all critical points of f are simple and their orbits do
not intersect. Although the reader can easily notice how to modify the considerations

to embrace the general case.

Consider the loops L,, 2 I10 U P 1, If there is a subsequence {Ln,} of loops with
even index at y then we come to a contradiction because this would mean there exists
an f "W -lifting 7,,, of Fn, déf [bij , bl]UI‘1,0 U[b2, (731,] which has endpoints 6,1,1. and Ci]. . On

the hand by Lemma C diam 7,,, S M’ - r,,, and dist(.§, J) 2 a - rn, which contradicts

39

to the assumption that c(Q,,,r,,,n) —> 0. Otherwise all indices of L,, at y are odd,
n 2 m. Consider L? = L,, — L,,, and observe that L? has even index at y for n 2 m.
Put Fn 2 [b1,, b1] U F1, U [b2,b3,], n 2 m. The fact that index of L1," at y is even shows
that there exists an f""-lifting 7,, of PM n 2 m, which has endpoints c1, and 63,. But
by Lemma C diam 7,, S M’-r,, and dist (§,J) 2 a - r,,, 6 E 7,,, n 2 m,just because
dist (5, J) 2 a’, 65 r. = [m1] u 12,, u [neg], n 2 m.

Thus the assumption c(Q,,,r,,,n) —-> 0 leads to a contradiction in any case and

this proves Theorem 1.4.8. I

All these results above served as illustration. Now Theorem 1.4.9 gives necessary
and sufficient condition for the localization property of A00( f) First we introduce

some notations:
Q, = {w E J : w — critical point and Cw = {w}},

92 = {w E J : w — critical point and Cu 75 {w}}.

With these notations we have the following definition.

Definition: Let f : C ——> C be a polynomial. We say that f is separated if there

exists 8 > 0 such that

dist (fk(w1),Cw,) > 8 for k E N, w, E 91 and we E 92.

Here is the characterization of the uniformly John property for Aoo( f ):

Theorem 1.4.9. Let f : C ——> C be semihyperbolic. Then Aoo(f) admits localiza—

tion if and only iff is separated semihyperbolic. A

40

Proof. Suppose f is not separated semihyperbolic. Then there exists w, E

91, we E 92 and a sequence {n;,};, such that
dist (f”"(w1), CW) —> 0 as k -> 00.

Without loss of generality we can assume that f"*(w1) —+ y where y E sz- Let
8 = 8(y,5/2) and C = 8 0 CW. Denote by Wk the component of f‘""(8) which
contains wl. Because w, is a critical point and f”*(w1) E CW2 these are at least two
disjoint components {C1}, of f ’m‘ (C) which are contained in Wk. The degree of the
map:

rum—>3

is D by Theorem B. Consequently diam C}, N diam Wk for i = 1,2 by Lemma C.

Also by Lemma C we have that for certain two disjoint components CLsz
dist (CLCE) S C(k) - diam Wk

where C(k) —>0as k —i 00.

Now it is easy to see that in this situation the uniformly John property is violated

(see Section 1.5). So localizability fails.

Conversely, suppose that f is separated semihyperbolic. We will show that Aoo( f)
is localizable. The idea is to consider two cases which are similar to the situation in
Theorem 1.4.2 respectively Theorem 1.4.4 above. First we introduce some notations
and prove a somewhat stronger version of the separation property. For w E 91, t > 0

and n E N we introduce:
T,°,”(t) = {C Q J :C is connected component of J, dist (f”(w),C) S t}

and denote

d1,“(t) = sup{diam C :C E T,‘,"(t)}.

41

With the above notations we have the following claim:

Claim: If f is separated semihyperbolic then there exists K > 0 such that for
t<€, wE91, kEN,

d,’f,(t) < K-t (1.4.1)

Proof of the Claim: Assume the statement is false. Then there exists w, E 91

and sequences {nk}k, {tk};, and {Ck}k, m, E N, tk > 0 and C], Q J such that

dist (f"*(w1),Ck) I
diam Ck < k for k e N.

 

Let us denote by t], the smallest integer (which exists by Lemma 3.3) such that
fe" (C1,) = Cw2 where wz E 92. Without loss of generality we can assume that we is
the same for each k. Since f is separated semihyperbolic there exists Ne E N such
that PM, (we) does not contain any critical values except for the ones situated on CW.
Then there is an ffk—lifting denoted by ng+No of PNO (we) such that C], C ng+No and
the map:

fe" 3 Ptk+1vo —> PNo(w2) is univalent.

By the distortion theorem for univalent maps we have

dist (3P5,+N0,Ck) dist (aPNeacwzl
diam C1, diam Cw,

 

 

=a>0,

where a does not depend on k. Thus f”*(w1) E ng+No if k is large enough. Again

by distortion theorem:

dist (f""(w1),C;,) dist (f”"+e"(w1),Cwe)
diam Ck diam Cw; .

 

 

This implies that dist (f""+f"(w1),C2) —> 0 which is a contradiction to the assumption

that the dynamics is separated semihyperbolic. The claim is proved. I

42

To continue the proof we introduce some notations:
17 = inf{diam Cfnee) :w E 92, n E N}.
By Lemma 1.3.3 we have that 77 > 0. For a reason which will become clear we choose

5 = min (4%,, g) . (1.4.2)

We also fix a large constant M, > 0 which will be determined later.

For x E J and r > 0 introduce:
n(r,x) = max{n : diam W,, 2 M1 or}, (1.4.3)

where W,, is the component of f‘k(8(f"(x),6)) which contains x. Denote by W,,,
the component of f‘k(8(f"(x), 6)) which contains f"”‘(x) and let y = f"("'”l(x). To
separate the cases similar to the ones in Theorem 1.4.2 and respectively Theorem

1.4.4, for r > 0 we introduce:
J2(r) = {x E J: W,,(mwc fl 9, = (l) for any k = 1,2,... ,n(r,x)},

and put J1(r) = J\J2(r).
Take first x E J1(r). This is the case similar to the one in Theorem 1.4.2.

Let us recall the topological disks U C U1 C U2 C f (U ) from Section 1.3. Denote
by 8,,(2), 8,,(2) the components of f‘"(U1) and f‘"(U2) respectively which contain
the point 2 E J. For 2 E J let us denote by N = N(z) the first integer n with the
property:

diam P,,(z) S diam C, + 6/2. (1.4.4)

Let us consider the open covering of J by the sets {pN(Z)}z€J. Since J is compact

we can choose a finite subcovering {PN,(2,)},:fi. There is an index ie E {1, . . . , L}

43

such that f"("’$)(x) = y E FN,0(2,0). We claim that in this first case (i.e., x E J1(r))
we have:

diam PN,0(Z,‘0) < 6/2.

To see this, recall that there exist w E 91 and k S n(r,x) such that dist (y,f’°(w)) <
6, diam PM, (2,0) S diam Cg,0 + 6/2, and therefore

g

dist (fk(w),Cz,.o) < 2.

By the condition (4.1) we obtain: diam C2,, S K- 36. Finally, we use (4.4) and (4.2)

to get:
. 6 5
diam PN,0(2,0) < (3K + 1) ° '2- 5,
which proves the claim.
Let us consider now the annulus AN, = PN,(2,) \ PN,(2,) and an analytic are

7, C AN, with the property that:
dist (71,8AN,) 2 6’ > 0 for i = 1,_L',
where 6’ > 0 is a fixed constant. Denote by 9, the topological disk with 39, = 7,, i =
1,. . . , L.
Let us summarize the chain of constructions:
(x,r) r—> n(r,x) —) y = f”(’"’”)(x) —> ie 2 ie(r,x) —> 9,00%).
Let 9,,(r) be the component of f '"(r’x)(9,o(,,,)) which contains x. Furthermore we

denote by U,(r) the component of f ""(r'x)(PN,o(,,,)(z,o(r,x))) which contains x. We

have that 9,,(r) Q Ux(r). Because diam PN.0(,_,1) (2,0(r,x)) < 5/2 the covering:

f'W) : U,(r) —> PN.

I0(r,1‘)

(2,0(r,x)) has degree S D,
by Theorem B.

We can now apply Lemma C to conclude that:

44
(a1) diam 9,,(r) ~ r

(a2) 9,,(r) is a’-thick at x for a’ = a’(f)

~

(a3) dist (J m (2,6), 012,07) ~ r.

As in the proof of Theorem 1.4.2 we define 9,,(r) = 9,,(r)\J. By [CJY] and by the
properties (a1), (a2), and (a3) we see that 9,(r) is the local John domain at x E J1(r)

with uniform John constant.

The second case is x E J2(r). The proof is similar to the one of Theorem 1.4.4.
We take x E .7: (r, M) (1 J2 and we are going to prove the statement of Lemma 1.4.3.
Namely we will show that for any M there are M1 (defining J2(r)) and M’, a > 0

independent of x and r such that
c(Q,r, M’) > a - r for x E f(r, M) 0 J2(r) (1.4.5)

If (1.4.5) is proved for all M S M(N, Me, D), where N, Me are from (1) of Theorem

2.2.2, then the localization follows by the algorithm described in Theorem 1.2.2.

Suppose (1.4.5) is false. Then there exist sequences {xk}k, {rk}k of points and
scales such that x], E F(rk, M) F] J2(rk) and c(xk,r,,,k) —> 0 as k —> 00. Let c,1,,c,2c be
centers of 9,1,,(rk), 92,,(rk) which are domains of prelocalization described in (1), (2),
(3) of Theorem 1.2.2. Let 7;, be a curve with diam 7;, < 2M . r,,. By our assumption
given any positive a for all sufficiently large k, there is no curve 7;, connecting c}, and
c2 with the properties:

diam 7;, S 2k - rk, (1.4.6)
dist (7],, J) 2 a ‘ T‘k. (1.4.7)

We obtain a contradiction and we will be done if we still construct 7,, with properties

(1.4.6) and (1.4.7).

45

Let us denote by n, = n(rk,xk) and y, = fem). Consider fixed small con-
stants 61,62 with 62 < 61 < 6. We denote by W,,2,W,,l and W], the components of
f‘""(B(yk, 52)), f'”"(B(yk, 61)) and f'”"(8(y,,, 6)) which contain xk. It is clear that
the map

f"" : W,, ——> B(yk,6) has degree S D.

For any k E N the disks W,,2,W,,1 and W,, are a’-thick at xk for some 01’ > 0. We

choose M1 large enough (depending on M) so that we have
8(xk,4M ° T1,)g W13.

Then 7;, Q W3, and we define F1, 2 f"*(7k), b}, = f”k(c},), i = 1,2. By Lemma C we
obtain that for any k

Cl ' 62 S diSt (b2, J) S 262,
for some c1: c1(f) > 0.

Then the curves I}, connect b}, and bi and I), Q 8(yk, 62). By the same reason as
in the proof of Lemma 1.2.3 we can find a curve P], Q B(yk, 6,) connecting b], and b:
with the property:
dist (Fk,J) > a where a = a(f) > 0.

Because x1, E J2(rk) the ball 8(yk, 6) is free from critical values coming from 91.

On the other hand:

diam (r, u r.) < 25, < n

by (1.4.2), and therefore the index of I", U P], with respect to critical values coming
from 92 is equal to zero. Consequently there is an f "k-lifting 7;, of F1, connecting c},
and Ci. Another application of Lemma C shows that 7;c satisfies (1.4.6) and (1.4.7).

This contradiction finishes the proof. I

46

Remark. Obviously critically finite polynomials are separated semihyperbolic and
thus A0,,(f) is uniformly John iff is critically finite. Still I think it’s illustrative to

see the independent proof in the criticlly finite case.

1.5 Uniformly John domains

Let us remind that p means the internal metric in a domain.

1.5.1 Definition. Domain 9 is called a uniformly John domain if for any two
points x1,x2 E 9 there exists a curve 7 = 7,1,3, connecting x, to me and lying in 9

which has two properties:

(i) V6 E 7 dist (6,39) 2 c1 dist (§,{x1,x2});

(ii) diam 7 S C2p(x1,x2).

1.5.2 Remark. Condition (i) is just the John type condition, that is the existence
of a “cigar” connecting x1 and x2 inside 9. Condition (ii) means that we can connect

x, to x2 by a path which is the best up to a constant. A

1.5.3 Theorem. A John domain is localizable if and only if it is uniformly

John. A

Proof. We have seen that in the simply connected case the localization holds. It
turns out that in this case uniformly John property holds as well. This is based on
the fact that we can choose 7 with the property (i) to be the hyperbolic geodesics (see
[NV] or [GHM]). On the other hand geodesics satisfy (ii) by a theorem of Gehring
and Hayman (see eg. in [P01]. So we are going to deal with non-simply connected

€3.86.

47

a) Let 9 be a localizable John domain and x1,x2 E 9. Let Q, be a point of 39
which is a closest point to x, and let r1 = [Q1 — x1]. If p(xl, x2) S %r1 one can choose
7 to be [x1,x2]. So assume that r = p(x1,x2) 2 -.:;r1. Let 9Q,(4r) be a local John
domain which contains the curves 7,,“, diam 7,,,,,,2 S 1.01r and [x1, Q1]. Let c be
a center of 9q,(4r) and 7‘, i = 1, 2, be two John arcs in 9q,(4r) connecting C with
x,, i = 1,2, respectively. Put 7 = 71 U 72. Property (i) follows immediately because

the John constants of local domains are uniformly bounded. As to (ii) we have a

simple estimate
diam 7 S diam 71 + diam 72 S 2 diam 9Q,(4r)

S 8Mr = 8Mp(x1,x2).

b) Let 9 be a uniformly John domain. First we use only the John property.
Applying Theorem 1.1.9 to a simply connected John domain 9 C 9 constructed in
[J02] we obtain a collection {96(r)} of simply connected John domains with John

constants uniformly bounded by Ce which satisfy the properties:

(1) diam 98(r) SMe-r, (2 1,...,N;

 

(2) U 9220") 2 Bp(Qa7');
(3) 95(r) m age) = a), for 1 ,e j.

As it was done several times before we introduce the control c(Q,r, M), M 2 Me,

and the sets f(r, M) = {Q : c(Q,r, M) > 0}.

Now we are going to prove the statement:

3 M’, a > 0 independent of Q and r: c(Q,r,M’) > a for any

Q E f(r,M).

48

Let us remind that this statement implies that 9 is localizable by the algorithm of

Theorem 1.2.2.

So let Q E .7: (r, M ), which gives a curve 7 Q 9 connecting centers 3,0" of
95(r),9g(r) respectively and such that diam 7 S 2Mr. Now it is time to use the
fact that 9 is a uniformly John domain. We know that p(c‘,cj) S diam 7 S 2Mr.

Using (i) and (ii) we obtain 7, diam 7 S 2C2Mr such that
dist (5, 39) Z ClT‘.
The statement above is proved with M’ = C2 - M and a = C1. I
1.5.4 Corollary. Let f be a polynomial. Let J do not split the plane. Assume

that either Cw = {w} or C, 74 {w} for all critical points w E J simultaneously. Then

the following assertions are equivalent:

(a) Aoo(f) is a John domain;

(b) A0,,(f) is a uniformly John domain. A

Proof. One needs to check only (a) => (b). By [CJY] (a) implies that f is
semihyperbolic. The application of Theorem 1.4.3 or 1.4.5 combined with Theorem
1.5.3 finishes the proof of (b). I

1.5.5 Corollary. Let f be a separated polynomial. Let J do not split the plane.

Then the following assertions are equivalent:

(a) A0,,(f) is a John domain;

(b) Aoo(f) is a uniformly John domain. A

49

1.5.6 Remark. It seems rather probable that uniformly John domains are Gro-
mov hyperbolic with the quasihyperbolic metric. At least due to discussions with
Juha Heinonen and Mario Bonk I strongly believe in this fact for Aoo( f) of separated

semihyperbolic f.

1.5.7 Remark. The condition (*) (that is “J does not split the plane”) is not
essential. One can replace components of J by components of filled-in Julia set K I

everywhere and get the same results for Aoo( f) = C\Kf.

1.6 Example of a semihyperbolic polynomial which
is not separated semihyperbolic

Let Ue,U1, U2 be topological disks such that U, C Ue, i = 1,2 and U1 0 U2 = (0.
Let f, : U, —-> Ue, i = 1,2 be branched coverings of degree 2. We can take f, to be
second degree polynomials with critical points w, and Julia sets J, Q U, for i = 1,2.
Choose f2 in such a way that w2 E J2 and orb(w2) is finite. Hence J2 is a dendrite.

Choose f, such that f,"(w1) —> 00 and therefore J1 is a Cantor set.

Using quasiconformal surgery we find a four degree polynomial f3 which is con-
jugated by a quasiconformal mapping cp to our dynamics. Two of the critical points
of f3 are escaping and the third one is in J3 with finite forward orbit. Let us denote

this critical point by we. It is clear that we 2 (p(w2).

Before we continue the construction let us state the following:

Claim: There exist ye E J3 and {nk};c Q N such that Cyo 2 {ye} and

dist (f;“(ye),Cw,)—>0 as k—>oo. A

Assume for the moment that the claim is true. Choose topological disks U3 Q U,

 

50

such that J3 Q U3 and f3 : U3 -—> U, is a branched covering of degree 4. Let us take
another topological disk U5 Q U, with U3 (1 U5 = 0 and a quadratic polynomial f4
which is a branched covering f4 : U5 —> U, with J4 C U5 and the critical w4 escaping.
More exactly we are going to choose f., in a way that f4(w4) = ye. Making another
surgery we obtain a polynomial f of degree 6 with three escaping critical points. Let

us denote by (I), and (122 the other two critical points which are in J = J(f).

If it) denotes the new quasiconformal conjugacy we have a), = w(w4) and (:22 =
1/2(w3). It is clear that orb (52,) Q w(U3) and since a, E tb(U5) we have that

dist (o1, orb (1.721)) > 0. Consequently f is semihyperbolic.

By our claim we obtain that
dist (f”"+l(w1),C,,—,2) ——> 0 as k —> 00,
which shows that f is not separated semihyperbolic.

This finishes the construction, we only need to prove the claim.

Proof of the Claim: Instead of f3, we, J3 we are going to use the notations f, w, J
and let V, W be two topological disks (corresponding to 90(U2), (p(Ue) in previous

notations) such that w E V r) J and

f : V —> W is a branched covering of degree two.

Also, without loss of generality we can assume that the escaping critical point of

f are in W\V. Fix xe E V and consider the sequence {11),},, of probability measures:

1
12:47 E 6,. (1.6.1)

31614030)

A well-known result of Lyubich [Lyl], [Ly2] and Mafié, Freire and Lopes [F LM] states

that {pk}k converges weakly to the measure of maximal entropy m of the polynomial

f.

 

51

As before let us denote by PN(w) the component of f‘N (V) which contains w.

Then the map
1"” : PN(w) -—> V is a branched covering of degree 2’”.

Using (1.6.1) we have

_ #{y E P~(W) = f"(y) = x0}
_ 41. .

 

#k(PN(w))
Put k = t - N and observe:
#{y E P~(w) =f1'N(y)= $0} = 2N ' #{y = f"‘”'N(y) = 330}-

Consequently we have:

N, . (t-l)-N : x0
ue-~(P~(w)) = 2 #{y . f4” (31) } z @117

 

Because pew —> m weakly we obtain that

m(PN(w)) = 2% for N E N. (1.6.2)

If we change w to any x E J we obtain similarly:

1

m(PNoD s ,..

for N E N and x E J. (1.6.3)

Let us denote by J, = {x E J : C, = {x}} and J2 = J\J1. A consequence of (1.6.3)
and Corollary 1.3.4 is that m(J2) = 0. By Birkhoff’s ergodic theorem there exists a

set XN Q J such that m(XN) = 1 and for any y E XN we have:

“m #{k = k S n.f’°(y) E P~(w)} = L. (1.64)

n—ioo n 2N

 

Since PN+1(w) Q PN(w) we obtain XN+1 Q XN. Let X“ = flNeNXN, then m(X“) =
1 is not empty. Let ye E X“ (1 J1. Then Cy0 = {ye}. On the other hand Ce 2

flNeN PN(w) and for y 2 ye in (1.6.4) we obtain a subsequence {n;,},, such that

dist (f""(ye),C,,,) —> 0 as k —) oo.

52

We are done. I

I should mention at this point that one can do a similar surgery to construct
an example of totally disconnected Julia set containing a nonrecurrent critical point.
The orbit of this point is infinite and hence we obtain a separated semihyperbolic
dynamical system which is not critically finite. I leave this constuction as an exercise

to the reader.

In the final section we prove a dynamical property of J. Namely, we are going to
show that a J is a boundedly finite-to—one factor of the one sided shift space. Our
main result: Theorem 1.7.3 will be used in the next chapter as we sudy equilibrium

states on for Hblder contionuous potentials on J.

1.7 Generalized polynomial-like maps. Geometric
coding tree.

Throughout this section (f, V, U) is a generalized polynomial-like system (see the

introduction), J = J I. Let us remind an important notion from [PS] and [Pr2]. One

constructs the geometric coding tree as follows. Let ze E U \ J. Let 2‘, . . . , 2“ be its
preimages. Let 7i be curves joining 2e to 2’, j = 1,... ,d, such that
d

 

orb(c) F) U 7i = (l)

'=l

for any c E Crit(f).

Let Z = {1, . . . ,d}N be the one sided shift space with a denoting the shift to the

left, and pg be the standard metric on E:

102(0fi) = 844”)

53

where k(a,8) is the least integer n for which 01,, # 8,,. For each sequence a we
put 71(0) 2 7"". Suppose that for every m, 1 S m S n and all a E Z the curves
7m(a) : [0, 1] —> U are already defined in such a way that f(7m(a)) = 7""‘l(o(a)) and
7m(oz)(0) = 7m“l(a)(1). Define 7"“(0) by taking respective preimages of 7"(o(a)).
Put 2,,(a) = 7”(a)(1). The graph T = T(ze,71,...,7“) with vertices ze, 2,,(a) and
edges 7"(a) is called a geometric coding tree with root at 2e. Given a E Z, the
subgraph composed by 2e,z,,(a),7"(oz) is called a—branch and is denoted by b(a).
The branch b(a) is called strongly convergent if {7"(a)} converges to a point as

n—+oo.

It is easily seen from Theorem B. of Section 1.3 that the following simple propo-

sition holds

Lemma 1.7.1 Let f be semihyperbolic, then there exist 0, < 00, 0 < 01 < 1,
such that

diam 7”(a) S 630?, a E Z. A

In particular each b(oz) strongly converges to a point of J. In general strong
convergence of b(a) holds for all 0 except a set of zero Hausdorff dimension in (2, pg).

Let n(a) = lim 2,,(a) be a point of convergence of b(a). Let r,,(a) {Te—f |z,,(a) — n(a)|.

Lemma 1.7.2. Let f be semihyperbolic. Then

#{zn((3) = l3 E Z, lznw) - 7r(Ot)| S k - 731(0)} S C(k,f).

for any a E Z and any n. A

Proof. Put x = n(a). Choose Me in such a way that

0,499“ 5
< _7
1—0, 2

 

54

where 5 is from Theorem B of Section 1.3 .

Denote 8 = on‘M°(a). Then 7r(8) = f"_M°(x). Let b),(a) = Uizk 7’(oz). Us-

Mo
ing (4.3) we have that diam bM0(8) S % S 5/2. Therefore bM(8) Q Ue dzef

1—9,
B(fn‘M°(x), g). Denote by W,,.Mo the component of f‘("'M°)(8(f"-M°(x), §)) which
contains x. We conclude that b,,(a) Q W,,-Mo, thus 2,,(a) E W,,-Mo, and so diam
W,,-Mo 2 r,,(a). Also f”"M° : W,,_M0 ——> Ue is a branched covering of degree at
most D (see Theorem B) and the same is true with 2Ue if we replace W -Mo by a

corresponding component of f ‘(n‘M°)(2Ue). Applying Lemma C from Section 1.4 we

see that W,,-Mo is r-thick at x, that is W,,.M0 D 8(x,r - r,,(a)).

Our purpose now is to enlarge W -Mo to the size of 8(x,k - r,,(a)). To do this
consider U, = 8(f""M°“M(x),§-) for a certain M to be chosen later. Let U2 be a
component of f ‘M (Ue) containing f "‘M0 "M (x) Denote by W,,-MO_M the component
of f‘("“M°‘M)(U1) which contains x. We have the covering f"‘M°‘M : W,,-M0_M ——)
U, which sends W,,- Mo to U2. This is the covering of degree at most D (see Theorem
B in Section 1.3.) and the same is true with 2U, with W,,-M0_M replaced by a
corresponding component of f ‘(n‘M0 "M )(2U1). Applying Lemma C from Section 1.4.

or using [HR] we see that with constants independent of n we have

diam W,,_M0_M diam U,
N

 

 

diam W,,.M0 diam U2.

But from Theorem B it follows that we can make diam U2 as small as we wish by
choosing M large. Since W,,-M0_M is r-thick at x given k we can choose M = M (k, f)
so large that

8(x,k - r,,(a)) c W,,_M,_M.

The degree of the map f"‘M°‘M : W,,_M,,_M —> U, is bounded by D independent of

n. So

#{Zn(fl) 1211(5) 6 8(x,k ' 731(0)} S D{2M+Mo(fl) I ZM+M0(3) 6 U1}

55

g D-dM+M°=C(k,f). I

Now we are in a position to prove

Theorem 1.7.3. Let f be semihyperbolic. Then 7r : Z —> J has the following

properties:

1) if is Holder continuous in metrics pg,pg,
2) 1r is onto;

3) #{u‘l($)} S K(f) for any x E J. A

Proof. Hélder continuity is obvious from Lemma 1.7.1. Also 3) follows immedi-

ately from Lemma 1.7.2. “Onto” part is also easy.

One just applies the criterion of accessibility obtained by Przytycki in [Pr2]. Or
one can proceed as follows. Given Q E J let 2,,(k)(a;,) —> Q. We may think that
a), —> a in Z. By Lemma 1.7.1 |2,,(k)(oq,) — n(ak)| S C1011“). Thus [Q — n(a)| S
|Q — 2,,(k)(ak)| + [2,,(k)(oz;,) — 7r(m,)l + |7r(oz;,) — 7r(a)| —> 0, when k —) 00. So Q = 7r(a)

and the proof is completed. I

Lemma 1.7.4. Iff is semihyperbolic then

dist(2,,(a), J)

r,,(oz) NI

 

where constants depend only on f. A

This is another standard application of Theorem B and Lemma C (see also [HR]).

 

Chapter 2

Invariant harmonic measure and
conformal maximality

2.0. Introduction

In this chapter we come back to our original problem of relating the harmonic measure

and the measure of maximal entropy for a GPL (f, U, V).

As it was explained in the introduction, we understand this relation as a problem
of rigidity. Namely we would like to prove that if we have a GPL (f, U, V) such that
m f z w, then (f, U, V) is conformally maximal i.e it is conformally conjugated to a

GPL (g, U', V’) such that mg 2 wg .

In this chapter we prove this for semihyperbolic (f, U, V) wich have a totally dis-
connected Julia set. Also here we construct a dynamical counterpart for the harmonic

measure.

In Section 2.1 we prove a necessary and sufficient condition for conformal maxi—
mality. This condition is the existence of an automorphic harmonic function, i.e. a
non-negative subharmonic function r on U, which is positive and harmonic in U \ J,

vanishes on J and satisfies

r(fz) = dr(z), z E V.
56

57

The proof of this characterization of conformality is essentially a combination of ideas

from [LyV] ,and [Lo].

Next we prove the result of conformal maximality for the case of PL with connected
Julia set. In this situation via the Riemann mapping theorem our result will become

a rigidity theorem of Shub and Sullivan in [SS].

This obsevation emphasieses that our problem is a rigidity problem of the har-

monic measure in the case of: - totally disconnected sets.

In Section 2.3 we consider the problem of constructing equilibrium states for
Hélder continuous potentials and semihyperbolic maps. This problem has been con-
sidered without the semihyperbolicity assumption by Denker and Urbanski in [DU]

and by Przytycki in [Pr3].

The problem in this generality has a solution if we assume a condition on the
topological pressure of our potential. We give a theorem which shows that for semi-
hyperbolic maps this condition is not necessary , proving that the semihyper bolic

maps are similar to the hyperbolic ones as far as ergodic theory is concerned.

In Section 2.4 we recall first the result of Hinkkanen , Eremenko and Mafié — da
Rocha in [Hi] , [E] , and [MR] on the uniformly perfectness of the Julia set. Then
a theorem of Volberg on the boundary behavior of ratios of harmonic functions is

applicable.

Furthermore a lemma of Grishin will give us that the Jacobian of the harmonic
measure is Hfilder continuous and we can apply the results developped in Section 2.3

to construct an invariant harmonic measure.

In the last section we solve our initial problem on conformal maximality. If we

58

assume that w z m starting with the invariant harmonic measure we are led to the
homologous equation:

long—logdzuof—u.

, This equation will allow us to construct the automorphic harmonic function which is
sufficient for the conformal maximality. This construcion of the automorphic function

is done according to a scheme in [LyV] and [V01],[Vo2].

2.1 A criterion for conformal maximality

In this section we give a characterization of conformal maximality that will be used

in Section 2.5. Namely we have
Theorem 2.1.1 Let (f, V,U) be a GPL system. Two assertions are equivalent:

1) (f, V, U) is conformally maximal;

2) there exists a non-negative subharmonic function r on U, which is positive and

harmonic in U \ Kf, vanishes on K} and satisfies

T(f2) = 617(2) A (Aut)

Remark I would like to mention that the direction 2) => 1) is due to Alexander

Volberg.

Proof of Theorem 2.1.2

2) => 1). Let T exist. WLOG the period of 7' over a simple closed contour

C C U \ V equals 27rd:

3r
(Ar)(Kf) — C fids — 27rd.

59

We define 1p in U \ V by
99(2) : 6(T+ii)(t)

It is a well defined holomorphic function. In fact,

/ 2213 (Ar)(V, n K!) = 3,130“ 0 f)(V.' 0 K1)
av,- n
d: , di I
= J(ATXfU/i f) 1‘3) = 3(AT)(1\I) = 27rd,,

where

d,- = deg(f l 14-).
Let us extend (p from U \ V into C\ V as a C 00 mapping with the only restrictions:
r(2)=zd, 2~oo; r740-

Pull back the standard conformal structure Co on (p(C \ V) and define

{game on C\V,
0':

0e on V.

Clearly 0‘ = 0e on U (not only on VI).

Measurable Riemann mapping theorem now gives a quasiconformal h, h(2) ~

2, 2 —> 00, such that
h*0’0 = 0' (2.1)
We define g = h"1 0 f0 h, V, = h‘1(V), U, = h‘1(U), and thus h : U9 —> U is

conformal. Let us check that g is a maximal system, i.e. that
wg = mg. (2.2)

To this end consider

T (2) a_er_l_ 108 l‘POhKZ), 2 E C\ V9;
9 d (roh)(z), z 5 U9.

60

First of all the definition is correct as by construction 7 = log [cpl on U \ V. The second
line of definition represents clearly a harmonic function in Ug \ K g and subharmonic
in U9. But also the first line represents a harmonic function (now in C \ V,,). This is
because (see (2.1.1))

(tpoh)"oe=oe on C\Vg

and 1p 75 0. Automatically r9 | Kg = 0 and 79(2) ~ log [2], 2 —> 00. Thus r9 is Green’s
function G, of A0,,(g) = C \ Kg with pole at infinity. On the other hand 0'9 2 r9 is

automorphic:

09(92) = ng(z). z e u. (2.3)
because T was assumed satisfying (Aut).

Now (2.1.3) implies that wg 2 AG,, has the following property
wg(g(E)) = A(Gg 0 g)(E) = d“19(5) (2-4)

for any Borel E C J9 such that g : E —) g(E) is injective. But as was explained
in Section 0 there exists unique measure satisfying (2.1.4), that is the measure of

maximal entropy mg. So w_,, = mg and we are done. I

1) => 2). We modify the idea of [L0]. Let us have a conformally maximal
system. We need to prove the existence of an automorphic function T. To do this
we may assume that our system (g, V, U) is already maximal (clearly the existence of
automorphic function is conformally invariant). Let {x,(€) 1;, denote all g-preimages
ofE E Jg counting with multiplicity. Let u E C(Jg) and g : E —-) g(E) be injective.

Then the fact that Jacobian Jm equals d implies

d-I/gw)[uo(g|E)-1]dm=/Eudm

61

Putting $00 = — log d we rewrite this line as

(P:.m)(uxE) = / P..(uxE>dm = [E udm

Now m-almost all Jg can be covered by this kind of E’s (m has no point masses).

Thus (P; 0m)( u): fudm, that IS

fiu(x,(§))dm (€)=d/ udm
Jgizl J9

And maximality serves to claim that

/;u( e,(.§ dw(§:=d/Jgudw (2.5)

To use (2.1.5) let us introduce <I>( (=2) Lg, (M ,and F(2)= f fig-gag — dfjg M9 E

 

Hol(V \ Jg). Let us prove that F E Hol(Vg).z To this end choose a contour C in U\V.

Then for every n 2 0

[Cznr(z)dz = /, a<s>/C§’,1f)—;E—,"—f--df,dw(€>/C§?:

= [,Zomrdw -d Jg€”dw(€)=0

according to (2.1.5). We proved that the singularities of F are removable. In other

 

words

‘1’(g(2))g'(2) - d¢’(2) = A(Z) E Hol(V)-
As Green’s function G satisfies C(2) = f log |2 — 5 [6160(5) + const we rewrite this line
using the notation G" for 5836' = (I), and H’ for £H.

G'(9(Z))g'(2) - d(7(2) = H’(z),

for a certain real harmonic function H in V. As G and H are real valued we also get

V(Gog)—d-VG=VHandso

Gog—d-G=H+constd=efHe (2.6)

62

Two cases may occur: a) He E 0 in V, b) {2 E V : He(2) = 0} is locally a finite union
of real analytic curves. If the first case occurs we got G as our harmonic automorphic

function. So let us consider b).

Let N be a neighborhood of J9, N C V, put F = {2 E N : He(2) = 0}. Then
F 2 UL, F,, where each I‘, is a real analytic arc. Clearly F covers J_,, (as He I Jg =
(G o g — G) | J_, = 0). As Jg has no isolated points we can throw away those F,
for which #(Jg 0 Pg) < 00. After this operation the rest of P), will cover J9. So
let J, C Fe 2 U1; F, and #(J9 f) F,) = 00, i = 1,...,m. Now it is clear that

g_l(P0) C F0.

We call a cross-point any point of Jg which is an intersection of two different arcs
F,, i = 1,. . .,m. If pe is a cross-point then the set g'“(pe) consists of cross-points
(as g‘ll‘ e C Fe). But the number of cross-points is obviously finite. So there is no

cross points at all.

Let 0e be a thin neighborhood of Fe in which a holomorphic symmetry 2 —> 2*

with respect to F e is defined. In 01 = g‘IOe we then get
92" = (92)"- (2-7)

Let us put C(z) = C(2) + G(2*), He(2) = He(2) + He(2*). Then (2.1.6), (2.1.7)
give us
(5 o g — d . 5x.) = 1310(2), 2 5 0,.
By definition 8e E 0 on Fe. But also this function is symmetric with respect to Fe
and so 38%“ E 0 on Fe. Thus 8e E 0 on (91 and we have a neighborhood 0, of J, in
which
Gogzd-C. (2.8)

Then a standard extension “by means of equation” gives us C on the whole V with

the same automorphic property (2.1.8).

63

Theorem 2.1.1 is completely proved. I

2.2 Conformal maximality in the connected case

This section has an illustrative purpose. We will show that in the case of a PL
f : U —> V assuming that the Julia set J is connected the result on conformal
maximality will basically become a theorem of Shub and Sullivan in [SS]. Let us

recall this result:

Theorem A Let F : T —«) T be an expanding analytic endomorphism of the unit
circle. Suppose that F is conjugated to the map 2 —> 2“ by an absolutely continuous
homeomorphism of the unit Circle. Then F is conjugated to 2 —) 2“ by an analytic

map (M6bius transformation of the circle). A

We also introduce here the main ideas and notations of the upcoming sections.

This machinery is called the Sinai — Bowen - Ruelle thermodynamic formalism.

Here is the result in the connected case:

Theorem 2.2.1 Let U and V be to topological discs , U C V and f : U —> V be
a branched cover of degree d. Suppose that we have w z m . Then f is conformally

maximal. A

Proof: Let us denote by K, the filled Julia set of f that is K, dzef 9.20 f-”(U).

Consider the Riemann mapping
R : C \ 1K0 ——> C \ID

where ID is the unit disk. Consider the annuli A1 = B(U \ Ky) , A2 = B(V \ Ky) and
themaszA1—>A2,F=RofoR'l.

64

By the Schwarz reflection principle we can extend the analytic map F across the
unit circle T so that now F is defined in annulus containing T. Moreover F (T) = T

and we get that F is an expanding analytic map of the circle.

By conformality of the Riemann mapping we have that (R‘l)*wp = w j. Also by
the construction of the maximal entropy measure in [Lyl] , [F LM] ( or see Section

1.6 ) we have that (R’l)*mp = my. Therefor we obtain that mp m we.

On the other hand we = t where t stands for the normalized Lebesgue measure

on the circle. Consequently we have that (1771}? = pdt for some p E L‘(d€).

Now let us note that

th'lozdoh

for a homeomorphism (actually quasisymmetry) h. Clearly h“ maps rm: to mza = t’.
So d(h'1)*t = dmp = pdt, i.e. h‘1 is absolutely continuous. In a similar way we

obtain that h is absolutely continuous.

So by the Theorem A , F and 2“ are conjugated on T by a real analytic map h | T.
This map can be extended as a conjugation of F and 20’ into a thin annulus around
T. From this thin annulus it extends automatically to A1. Thus F is conformally

maximal (even conformally equivalent to 2").

This conjugacy to z —+ 2‘1 may not be true for f. One the other hand we have a
”one sided conjugacy” which is enough for conformal maximality. To see this, observe
that without loss of generality we can think now that F (2) = 2".

For the function 2 —> z“

, the function r(z) = log(]2|) satisfies the property
(Aut) from Theorem 2.1.1. The function r, = r o R is the automorphic function

corresponding to f. Applying Theorem 2.1.1 we are done. I

Let us illustrate our approach using a simple example of (F, A2, A1), where A2 C

65

A2 C A, are topological annuli containing the unit circle T and F is a regular (1
covering F : A2 —+ A1, F(T) = T. Formally this is not a GPL because we map
annuli instead of discs. But the ideas are more conspicuous here because the Julia set
Jp = T and harmonic measure w = t = Lebesgue measure on T are so simple. Also

automatically F is expanding on T.

We quote now basic facts of Sinai-Bowen-Ruelle formalism for expanding endo-

morphisms. To begin with let us recall some facts. The pressure of function 90 E C (T)

is denoted by P(F, (,9) (see [DGS] or [Wa] for properties) and is defined by:

P(f, cp) =sup{h,,(F) + chde/ : z/ is F—invariant ergodic probability measure} By

h, we denote the entropy of the measure V.

Remind that th Perron - Frobenius — Ruelle operator P2 : C (T) —> C (T) is given

by Pew(x) = EFF, w(y)e""’(y) and L,,, : C(T) —> C(T) is defined by L,, = P,p_p(p,,,,).

Lemma 2.2.2. Suppose that F is expanding and cp is Holder continuous. Then 5,,
is almost periodic, i.e. for every w E C (T) the sequence of functionsfigw is bounded
and equicontinuous. There is a positive fixed point for L,,, namely a function the
such that Lethe = we. Moreover we = lim [.31 and it is Ho'lder continuous. There is a
unique probability measure 77 on T such that £317 = n. We also have £31,!) —> ’l/Jo'fT it) dn

uniformly as n —> 00. A

The expanding case is similar to the case of one-sided topological Markov chain
and one can find the proof of the lemma in [B0] , [Ru] ,[PP]. If our dynammics is not

expanding but semihyperbolic this result still holds as it is shown in the next section.

Given a measure V with good J acobian Lemma 2.2.2 allows to construct an equiv-

66

alent F -invariant measure 11 with good density 31%.

Lemma 2.2.3. Suppose that F is expanding and V is a probability measure on T

with Jacobian J,, such that
(,9 = — log J,, is Hblder continuous.
Then there exists a unique F-invariant measure 11 such that
log 311—:- is Hblder continuous.

Moreover if

w is real analytic on T

then

log j—H is real analytic on T. A
V

n—l

Proof. Let E,,(y) = 52:0 “my”. F is expanding we can easily write (xe is any

point on T)

. 1
P(F,s0)=,}gg;,-log Z En(y)

F”11:50

Taking sufficiently small neighborhood V of xe and using Hblder continuity of (p we

see that

E..(y) % 1
E,,(z)

for any two points y,2 in the same component V,, of f‘"(V). This and the fact

 

that (,0 = — log J,, imply 217%,sz E,,(y) S C 2 V(V,,) the sum being taken over all

components of f‘"(V). Now it is clear that P(F,<p) S 0.

On the other hand let us compute PgV. Fix a subset E C T on which F is

injective. Then (we repeat the reasoning on p. 116 of [P3])

Pie/XE) = fauna: / ewe-1.1.:

f(E)

67
= / ew°(f|E)‘1°fe-cpdu : V(E)
E
Decomposing T into a finite number of E, on each of which F is injective we get
P;V = V
Thus the spectral radius of P; is at least 1. But r(P,;) = 5131390) 2 1. So P(F,w) 2 0

which shows P(F, w) = 0.

Now Lemma 2.2.2. can be applied to w = —log J,, to obtain we 2 lim,,_,oo P31
and 17, P5117 = 17. By uniqueness 77 = V. Clearly 1i d=cf wedV is F-invariant and log we is

Hblder continuous because we is and we > 0.

If 90 is real analytic then we is real analytic and we are done. I

2.3 Hfilder continuous potentials and semihyper-
bolic dynamics.

The purpose of this section is to extend the results in Lemma 2.2.2. and Lemma 2.2.3
to the non-classical case of semihyperbolic GPL. It is clear that we can think that we

are dealing with polynomials.

In this section f : C —+ C denotes a semihyperbolic polynomial (or rational map)

and w : J —> IR is a Hblder continuous ”potential” on the Julia set J.

Using the notations in the previous section we prove first the following analog of

Lemma 2.2.2 :

Theorem 2.3.1. Suppose that f is semihyperbolic and w is Holder continuous.
Then £2 is almost periodic, i.e. for every w E C(J) the sequence offunctions £321)
is bounded and equicontinuous. There is a positive fixed point for £,,,, namely a

function we such that £,pwe = we. Moreover we = lim £31 and it is Hb'lder continuous.

68

There is a unique probability measure 7] on J such that £111) 2 n. We also have

£gw —> we - fJ wdn uniformly as n —) 00. A
As a consequence we obtain :
Theorem 2.3.2 The measure V = wen is the unique equilibrium state for w. A

Note that by functional analysis reasons there exists A > 0 and a probability

measure 17 such that:

From the proof of Theorem 2.3.1. it follows that log A = P and n from above must

be the same as in Theorem 2.3.1.

The above result has been proven by Denker , Urbanski in [DU] and independently
by Przytycki in [Pr3] for any rational function f but assuming the additional condition
: P( f, w) > sup J w. In [Pr3] there is an example showing that the condition P( f, w) >
sup J w is essential. In the example the map f was parabolic. Nevertheless we can

prove this result without the inequality assuming that the map is semihyperboic.

We are going to use ideas from [Pr3] but also a stronger version of Theorem B
from Chapter 1. To formulate this result let us denote by 8,,(x, 5) the component of
f‘”8(f"(x),5) containing x. Based on the Theorem B in Section 1.3. one can easily

prove:

Theorem B [ff is semihyperbolic, there exist so > 0, L, > a, > 0 and 0 < 0 < 1

such thatfoerJ,nENand5<5e :

diam8n(x,5) < Lleald"

69

We start with preparations following the scheme in [Pr3]. We prove first:

Lemma 2.3.3. There exists c > 0 such that for any x,y E J and n E N we have

Proof: Consider 5e > 0 from Theorem A and let {8(x,,52‘1)},=1,,,,,x be a finite
covering of J with x, E J. We fix i E {1, . . . , K} and we are going to prove the lemma

for x,y E 8(x,, 929).

Let us denote by:

E,,(z) = er(2)+...+.p(fn—1 z)

and with this notation we have:

Pguxx) = Z E.(z.)

z;Ef""x

and

P:(1)(y) = 23 E.<z.)

ZyEf-"y

Let us denote by {8,}12‘, the components of f‘”8(x,, 529). Then

Kn
P3(1)($) = Z Z En(ze)

J=l 21:68}

and

K.
P£(1)(y) = Z Z En(zy)

j=1 2968,

70

. Since we have that #{2x : 2,, E 8,} = #{23, : 2,, E 8,} we can consider the pairs

(2,,2y),2,,,2y E 8,.

In order to prove our estimate it is enough to show that

D]

n(z.) <6
En(zy) _ .

 

Without loss of generality we can assume that 50 < 1 and by Theorem A we obtain
that [szx — szy] S len'k which gives : |<p(f"z,,.) — w(f"2y)| S LL?0"(""‘) for

k = 0,1,. . . ,n — 1. This implies that:

E,,(z,)
En(zy)

a I
1 1-90

 

 

S eLL

which proves our estimate and thus the result for x, y E 8(x,, 911).

To finish the proof we need to show that for e.g.

P£(1)($1)

_— < C
P.I.‘(1)($:>.) _

. First observe that there exists an N E N depending only on 50 such that f ‘N x2 (1

8(x,, 9251) # (0. Let us take y E f’Nx2 fl 8(x,, 93), then:

P£(1)($1)<c
P£(1)(y) —

by our previous consideration.

On the other hand we have : P:(1)(y) S eNIIWIlP£+N(1)(x2), and therefore:

P£(1)($1)
P£+N(1)(~’v2) "

 

Furthermore we have : Pg(1)(x2) S dNeNllwlng+N(1)(x2) and consequently:

P£(1)($1) < C
P3(1)($2) —

The proof is finished. I

71

Lemma 2.3.4. For any x E J we have:

lim llog P;(1)(x) = P(f,w).

n—+oo n

Proof: The ineguality lim,,.,,,o filog Pg(1)(x) S P(f, w) is proven in [Pr3] for any

rational map f and any continuous function w.

For the opposite ineguality Przytycki used essentially the condition P(f,<p) >
sup (,0. We’ll see that because of the semihyperbolicity we don’t actually need this

condition.

Let us first recall some definitions from topological dynamics. We call two points
x,y E J to be (n,5) separated if there exists k E {0, . . . ,n} such that [f’°x—f"y| > 5.

A set F is (n,5) separated if any two points of F are (n,5) separated.

The pressure P = P(f,<p) of a continuous function w can be calculated by the
formula (see e.g [Wa] p. 209):

P 2: lim lim 1 log sup{Z En($)}a

5—10 n—ioo n
xEF

where the supremum is taken over all (n, 5) separated sets F C J.

By this way we see that for any 6 > 0 there exists C 2 C(6) and 5 = 5(6) such that
for any n E N we can find (n,5) separated set F(n) such that

Z E,,(z) 2 Clenlp-al.

xEF(n)

Without loss of generality we can assume that 5 < 5.} where 50 > 0 is from

Theorem B. Let us choose 51 , 0 < 51 < 5 such that for x E J , k E N we have

diamB)c < i; for any component 8), of f’k8(x,51).

 

72

This choice is possible by Theorem B. Let us choose a covering {8(x,, £21) 1;, of

J with x, E J , i E {1, . . . , K}. This covering will stay fixed in the remaining part of

the proof.

Pick n E N and consider the corresponding (n,5) separated set F (n) Introduce

the sets

2,0,) = {2 e F(n) : f”2 E B(flii, 5:51)},
foriE {1,....,I{}

Because U£1Z,(n) = F(n) we obtain that:

K
2 Z E,,(z) 2 Cle”(P-6).

i=1 2EZ,(n)

From here it follows that we can choose to E {1, . . . , K} such that

Z E,,(z) Z c2e”(P"’).

262,001)

Since F (n) is separated for each y E f‘”x,0 there is at most one 2 E Z,o(n) such

that z E 8,,(y) where 8,,(y) denotes the component of f"8(x,o, 521) which contains y.

Let us introduce the set:
Y(n) = {y E f_”x,0 : there exists 2 E Z,,,(n) such that 2 E E,,(y)},

and observe that for each 2 E Z,0(n) there is at least one y E Y(n) such that 2 E
Bn(y)-

Exactly as in Lemma 2.3.3. the Ht'ilder continuity of w and Theorem B imply that

E,,(z) S C3E,,(y) for 2 E 8,,(y) and therefore we can conclude that:

Z En(y) Z C4en(P—6)
116W")

73

where c., = C4(6),c4 = :13.

A consequence of this is that:

P£(l)(x,0) Z c4e”(P"’).

This estimate and the previous lemma imply that :
P:(l)(x) Z C5e”(P_"),
for every x E J and c5 = c5(6).
This implies that for any 6 > 0 we have:

lim llog P;(1)(x) _>_ P — 6

71—)00 n

for every x E J.

This proves the ineguality wee need to finish the proof of the lemma. I

Consider the operator P; conjugate to P2 . As mentoined earlier there exists a
number A > 0 and a probability measure 1] on J such that P117] = A17. The next result
is based on a standard calculation and its proof can be found in [Pr3] it proves the

uniqueness of A and gives information on 17.

Proposition C If Pgn = An then A = 5P, the measure 17 has Jacobian J,, = eP‘V’.
Any probability measure 7], which has Jacobian J,,1 = Ae"? will satisfy P5771 = A171,

(and thus A = 5”). A
Now we can give the:

Proof of Theorem 2.3.1.: We denote by: £ = P¢_P 2 {PW

For w , continuous on J we have that: L wdn = I] £(w)dn. Observe that for

n20:

Supl£"(¢)l S supltbl -sup|£"(1)l

74

On the other hand because I, £"dn = 1 from Lemma 2.3.3. it follows that {£”},,

is uniformly bounded.
Now let us prove the equicontinuity of {£n(w)},,.

Denote by 5 = Ix —- y] < 50 and applying Theorem B by the same argument as in

the proof of Lemma 2.3.3. we obtain:

53(1)(~’v)
53(1)(y)

U\

("D
2
(g

L L a
where c, = 1:5,), .

 

This estimate and the boundedness yields the equicontinuity of {£’,;,(1)},,.

A

Let us denote by E,,() the exponential sums associated to £". We have the

estimates:

l£$(¢)(w)—£$(¢)(y)l=l Z3 E.(z.)w(z.)— Z3 En(zy)¢(zy)ls
zyEf‘"y

z,Ef‘"x

S 2 Enllel¢(zx) _ Tbs/ll + I Z: [Enlzrl " En(zyllib(zy)l S

zIEf—"x z;E]‘"x

SH 5‘0) ll 'SUPer) - ¢(zy)|+ || «A ll '|£"(1)($) - £"(1)(y)|.
from where it follows that {£;(w)},, is equicontinuous.

The operator £ satisfies also another property: it is primitive i.e. for every w 2 0

,w 35 0 there exists n > 0 such that £(w) > 0.

Now we can refer to a general theorem about positive , almost periodic and prim-
itive operators ( see [Lyl],[Ly2] or [PP] ). For every such operator not contracting to

0 there exists a eigenfunction we for the simple eigenvalue 1.

75

We have £ 2 P, + P2 where P1 is the projetor to span we, ker P1 is invariant
under £ and £"(w) -—> 0 for any w E ker P1.

Because P1 = F - we for a continuous functional F and £ is 77 invariant (i.e.
fJ wdn = f, £(w)d77 ) we have that ker P1 = ker 17. We choose F such that F(l) = 1.
For every continuous w , £“(w) —) F (w) - we, therefore 1 = F (1) - fJ wedn and by our
choice of F(l) we have F(we) = n(we)) = 1, thus F = 17.

By the way this consideration proves the uniqueness of the probability measure 17

satisfying P517 = An. Remind that we also have A 2 5” by Proposoition C.

The Hfilder continuity of we is left as an exercise. I.

Corolarry 2.3.5 Let 171 be a probability measure such that (,0 = — log J,, is Holder

continuous. Then P(f, w) = 0. A

Proof : Observe that 17 has Jacobian J,, = 1 . 5”. By Proposition C.2. we have

that n is the solution of P; = 1 - 7]. Now also by Proposition C.2. we have that

P(f,w) = 0. I.

To prove Theorem 2.3.2. we need to use the result at the end of Chapter 1.
Namely we need here Theorem 1.7.3 saying that there is a boundedly finite - to - one,

Hblder continuous semiconjugacy II : Z —) J.

Proof of Theorem 2.3.2. The proof of the fact that V = wen is an equilibrium

state for 1,0 is the same as in [Pr3]. We repeat it here for the convenience.

First we show that V = wen is invariant under f: for a continuous function h we

can write:

n(wo - h) = 77(£3(t()o) ‘ h) = n(fiwo - h 0 f)) = n(ibo - (h 0 f))-

76

Next we estimate the entropy of V:

h,,(f) Z flogJudefloandV+/10g(weof)dV—/logwedV =
J J J J

=/logJ,,dV = P(f,w) —/(,0dV.
J J

Because the opposite ineguality is always true (see e.g. [Wa] p. 218) we have that V

is an equilibrium state.

The proof of the uniqueness of the equlibrium state is the more difficult part in
[Pr3] and it uses the ineguality P(cp, f) > sup go. We do not have this inequality but

we are going to use Theorem 1.7.3 instead.

To prove uniqueness let us suppose that V1 is an equilibrium state for w. There is
a standard way (see [Bo] p. 91) to construct a o invariant probability measure 11, on

2 such that x,y, = V1 .

It is clear that hp,(o) Z h,,,(f) (see e.g [Wa] p. 89). On the other hand the
semiconjugacy 7r : Z ——> J is boundedly finite - to - one (see Theorem 1.7.3.) and

thus we can apply a theorem of Ledrappier and Walters in [LW] to obtain that in fact
hu1(0) = hu1(f)'
Furthermore the function (,9 0 7r is Hdlder continuous on 2 ( Theorem 1.7.3.) and

we can apply the well known theory on the shift space 2 saying that the potential

4;) 0 7r has a unique equilibrium state pe. Consequently:

P(a.«,oovr)=h..(a)+/ SOOWdfloZ
)3

h..<o-)+ / eondu1=ht<a>+ / rordvl=P(f,r)-
Z 2:

Consider the measure Ve = rune. Using again that 1r is boundedly finite - to - one,

77

obtain that h,,0( f) = h,,0(o). Therefore we can write:

P(o,<p 0 7r) 2 h,,0(a) +/

r 0 rduo = h..,(f) + / rdVo S P(f.r)-
:3 J

Thus we have the equality P(o, w 0 7r) 2 P(f, w).

This means that P(o,cp 0 7r) 2 hu,(o) + f: w o 7rdp, which implies that 111 = he

by the uniqueness of #0-

It follows that V1 2 r,,pe and by the same argument V = rune also and we are

done. I

Corolary 2.3.6. Let n be a probability measure on J such that w = —log J,, is
a Hb'lder continuous function on J. Then there is a unique equilibrium state V for (,0
such that n m V . Moreover £3 = we where we is a Hb'lder continuous function on J

and we have the relation:

h,,( f) = /, log J,dV.

Proof: By Proposition C we have that n is the solution of P111] 2 17 and by
Corolary 2.3.5. P(f,w) = 0. Applying Theorem2.3.1. obtain a H61der continuous
function we > 0 such that Potbo = we. By Theorem 2.3.2. the measure V = wen

is the unique equilibrium state for (,0. Because P(f,w) = 0 we obtain also that

h,,(f) = leoandV. I

2.4 Invariant harmonic measure with good den-
sity

In this section we consider a semihyperbolic GPL (f, U, V) with totally disconnected

Julia set. We show that in this situation the harmonic measure has an invariant

78

counterpart.

The way to proceed is to study the boundary behavior of harmonic functions
in uniformly John domains. In addition to the localization property proven in the
previous chapter, we also use that the Julia set is uniformly perfect. This notion
was introduced by Pommerenke in [P02] meaning a certain thickness in the sense of
potential theory. Let us remind that the set E is called uniformly perfect with UP
constant a > 0 if

cap(8(x, r) 0 E) Z a - cap(8(x, r)).

For Julia sets uniformly perfectness was proved independently by Hinkannen in [Hi],

Eremenko in [E] and Marie with da Rocha in [MR].

Let 9 be a uniformly John domain with uniformly perfect boundary and denote by
pg the internal metric in 9. The next result is due to Alexander Volberg (see [BVl],
[BV2]), and it will be essential for us. It is called the Boundary Harnack Principle
(BHP):

Theorem D Let u,v be two positive , harmonic functions in 9 and vanishing on
Uefl39 where Ue is a topological disk. There exists 5 > 0 such that for any topological
disk U, with U, C Ue there is a constant K = Keep, such that:

u(:6) , u(y)
v(II?) ' v(y)

 

 

 

— 1' S K- [pe(x,y)]‘ for x,y E U1 ()9

This result is a generalization of a result of Jerison and Kenig in the case of
nontangentially accessible domains. By our main result in Chapter 1 our domain

9 = U \ J is a uniformly John domain with a uniformly perfect boundary. Moreover

79

we can easily see that there is constant C, > 0 such that:

1

3 (x,y) S la: - yl S Cxp(:v,y)
1

This equivalence of the two metrics implies the following ”ratio condition” (R) on

Green’s function G of the domain Aoo(f) = C \ .ll:

C(fél) , G(féz)

30.e>0V€1,€2€V\Kf’ C(61) ' G(£2)

 

— 1 S Clél — Czle- (R)

To see how powerful is (R) let us use the following version of A.F. Grishin’s lemma

[Gr]:

Lemma E Let V be a neighborhood of a compact K and let u,v be two subhar-
monic functions on V, vanishing on K and positive and harmonic in V \ K. Suppose

that for every x E 3K there exists the limit

E-w v(x
eeV\K

and this limit is continuous. Then due 2 pdpv, where umpv denote the Riesz mea-

sures ofu and v. A

Now we put it = G o f,v = G. Then clearly

He(E) = W(f(E)) (2-4-1)

for each Borel E C Jf such that f : E —-+ f(E) is injective.

For x E J I denote

C(ffb‘) a_:1_ “m logGUO.

”0“) = -10, Go) 6(a)

(eV\K,

 

80

By (R) the limit always exists and represents a Holder continuous function cp.

Now the Lemma E and (2.4.1.) show

was» = / e-Wwo)

E

for any Borel set E such that f : E ——> f(E) is injective.

Corollary 2.4.1 Let (f, U, V) be a semihyperbolic CPL with totally disconnected
Julia set. Then harmonic measure has the Jacobian Jw such that log Jw is Ho'lder

continuous. A

We have made the necessary preparations and now we are in the position to apply

the results of the previous section to conclude:

Corollary 2.4.2. Let (f, U, V) be a semihyperbolic GPL with totally disconnected
Julia set. Then there is an invariant measure )1 such that ,u m w. Moreover, we have

that the logarithm of the density: 1P0 = $5 is Holder continuous. A

The measure )1 is called the invariant harmonic measure.

2.5 Homologous equation and conformal maximal-
ity

The purpose of this section to prove our main result on conformal maximality. We are
going to use the characterization of conformal maximality given in Theorem 2.1.1.,

so our goal is to construct an automorphic function r. The construction is based on

ideas from [LyV],[Vol] and [V02]. Here is our final result:

Theorem 2.5.1. Let (f, U, V) be a semihyperbolic GPL with totally disconnected

Julia set. Ifw z m then f is conformally maximal. A

81

Proof: The setting is as follows. Given (R) we have already proved that (p(x) d:ef

log 50533 dzef limb”, log %5 is Holder continuous and there exists rho = e’7 (the eigen-

vector of RP), with Holder continuous *7 and such that do - w = u.

By the uniqueness of equilibrium states (Theorem 2.3.2) we get that u is ergodic

and hence m = )1. As a consequence:

Having in mind that JW 2 e'”, Jm = d we rewrite (2.5.1) as follows (7 = log tho)
90(1‘) +10gd = 7 0 f(x) - v(x), 11: E '11- (2-5-2)

This is the homologous equation connected to our problem and the all construction

is based on it. Having (R) in mind we can notice also that

G
log fl — logd + 7(f(:1:)) — 7(a) S clz — :cl‘. (2.5.3.)
C(Z)
From our considerations it will be clear that there is no loss of generality to assume
that there is a repelling fixed point p of f which is in J. Denote by B a small disc

centered at p such that all components B, of f ‘"B containing p are in B and B is

free from critical points of f.

We are going to build the automorphic r in three steps.
Step I: construction of r on B

Let us denote by g the inverse branch of f’1 : B -> B; C B. Now (2.5.3.) will

give:

—— < n o. O
llG( ) 1| Cq, (254)

82

for some q < 1.

Now (2.5.4) implies that the following limit represent a subharmonic function

r1(z) : lim d"G(g"(z)),z 6 B (2.5.5)

n—hw
and 7'1 is harmonic in B \ J.

Notice that 7'1 is automorphic on B1: 71(fz) = dr1(z). Furthermore by (2.5.2)

and (2.5.3), recalling that (p(x) = lim z—n log 596% we obtain the following

z€B\J

(?(Z)

.23(J‘T‘(Z)

 

= eu<x>+i<Ph a: e B n J (2.5.6)

Let us consider the union of backward orbits of B : O = Unzof‘”B.
Step II: extension of r to 0

Let Bo be a component of f ‘"B for some n > 0. We define a function r3 on B9
by:
1
792(2) = d—nn(f"z), z 6 Ba (2.5.7.)

We would like to prove that re2 does not depend on 0 (and n).

First notice that (2.5.2.) and (2.5.6.) imply that :

(i(Z)
213.1} 2
1689\J T9 (2)

= e-MWP), a: 6 Ba 0 J (2.5.8)

 

Therefore the above limit does not depend on 0.

It follows that for any two branch 01, 02 we have that

. 73 (Z)
131} —2‘—=1, (1:639,ntsz
168910892\J T62(Z)

83

Now we can apply Grishin’s lemma (Lemma E from the previous section ) to
obtain that Ar:l = ATE?2 on B9l 0 B92 and hence the function r92l — T02; is harmonic in

B91 0 B92 and it vanishes on B91 0 B92 0 J.

Now , either r31 E r921 or B91 0 B92 DJ is covered by a finite number of real analytic
curves. It’s not hard to see that if the latter happens the whole J can be covered by
a finite number of real analytic curves so this will be the case for any pair of 61,02

for which B91 0 B92 is not empty.

Furthermore , as in the proof of Theorem 2.1.1. we can consider the situation
when the curves are disjoint. Now let * be a holomorphic symmetry with respect to

these curves. Instead of re2 we are going to work with
def 1:
73(2) = 73(2) + 792(2 )-

The advantage is that now r31 E r32 in B91 0 B9,.

In an case we obtain a function 7'4 on 0 such that 7'4 = r3 or 7'2 if the first
y |Bg o a

possibility “7921 = r922” always occurs).

It is clear that our function T4 has the automorphic property on f ‘10.

Since J is totally disconnected there is a number N > 0 such that f "N U C B. In

the last step we are going to extend r4 to the whole U.
Step III: extension of r to U

Consider z E U which is not a critical value of f N . Choose a topological disc free
from critical values of f” and containing both 2 and p (here we assume without loss

of generality that p is not a critical value of f N )

Let VN, be the component of of f ‘N V, containing the point p and contained in

84

B. Then the map f"N : V —+ VN is univalent and we can define:
75(z) = dNr4(f'N(z)), z E V

It is clear that 1'5 does not depend on V since TISVnB = 7'“. We extend r5 to the critical
values of fN by continuity.

5

Because 7' is a positive harmonic function, harmonic on U \ J and vanishing on

J we only need to check the automorphic property.

To do that let us denote by Bl an arbitrary component of f‘lB. We are going

4

to show that TISBl = r . Since 7'4

was automorphic on B1 this proves that r5 is

automorphic on V,- where V,- contains B1.

Let us pick 2 6 B1 and an appropriate univalent branch fe’N of f ‘N . Put 21 =
fol-N2 and by our definition we have 75(2) = dNr4(zl). On the other hand observe

that fN+lzl 2 f2 6 B.

By the automorphic property of r4 we have:

1 1
74(21): dN+lT4(fN+121) = W740”

 

It follows that r5(z) = §r4( f z) and consequently, by the definition of r4 in Step II
we have: r5(z) = 74(2) for z 6 B1. This shows that TISV. is automorphic. As Bl was

arbitrary we obtain Tfi,‘ is automorphic for any i.

This concludes our construction and proves the theorem. I

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