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Rigidity and Dimension of the Harmonic Measure

on Julia Sets

By

Irina Popovz'cz'

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics
1998

Prof. Alexander Volberg

ABSTRACT

Rigidity and Dimension of the Harmonic Measure

on Julia Sets
By

Irina. Popovz'cz'

The harmonic measure in a dynamical context appeared for the first time in a
paper of Brolin, where it was established that the harmonic measure w associated
with the unbounded component of the complement of Julia set for a polynomial is
equal to the measure of maximal entropy.

The comparisson of these measures turns out to be very helpful in understanding
the generalized polynomial like systems (GPL). Douady and Hubbard have proved
that such GPL are quasi-conformally conjugated to polynomials. The first part of
my thesis contains the proof of a necessary and sufficient condition for a polynomial
like system to be conformally conjugated to a polynomial and a necessary condition
for GPL. It also contains the proof of existence of invariant harmoinc measures for
GPL.

The final part of the thesis is related to a problem that goes back to Carleson
and to P. Jones, T. Wolf and N. Makarov, of comparing the Hausdorff dimension of

the harmonic measure on a compact K and the Hausdorff dimension of the set K

itself. It has been conjectured (A. Volberg) that for all disconnected Julia sets J the
harmonic measure has dimension smaller than J.

The second chapter of the thesis contains the proof of a Boundary Harnack Prin-
ciple for Denjoy domains whose boundaries are uniformly perfect.

This result is used in the final chapter where it is proved that A. Volberg’s con-

jecture is true for the Julia sets of Blaschke products with one parabolic point.

ACKNOWLEDGMENTS

I am very fortunate to have Professor A. Volberg as my advisor; I am deeply
grateful to him for his continued support, kindness and graceful guidance without
which my research would not have been possible. He introduced me to the field of
Dynamical Systems and reshaped the way I think in Complex Analysis.

It was a pleasure to have been a student of Prof. Anca Maria Precupanu and
Prof. Teodor Precupanu at Alexandru Ioan Cuza University in Iasi, who taught me
a lot and who encouraged me to pursue research in Analysis.

I am indebted to the professors in the Mathematics Department at Michigan State
University for the wonderful years I spent here. I wish to thank Dr. Graczyk, Dr.
Nazarov and Prof. Newhouse for hours of discussions and advice, also Prof. W. Sledd
and Dr. S. Treil.

I am also grateful to Z. Balogh, M. Denker, F. Przytycky, M. Urbanski and many
others for valuable suggestions. I have also learned a lot, mostly indirectly, from
Professors L. Carleson, M. Lyubich and N. Makarov.

While working at my disertation I had the opportunity to meet a number of
mathematicians who encouraged me and showed interest in my work: K. Astala, A.
Eremenko, J. Heinonen, A. Hinkkanen, S. Rhode, M. Zinsmeister. I would like to
thank them all.

iv

TABLE OF CONTENTS

INTRODUCTION 1
1 Generalized Polynomial Like Systems 9
1.1 Necessary and Sufficient Conditions for Conformal Conjugation . . . 9

1.2 Construction of Invariant Harmonic Measure and of the Automorphic
Function .................................. 15
2 Boundary Harnack Principle 22
2.1 The Main Lemma. ............................ 22
2.2 Harmonic Rigidity. ............................ 29
3 Construction of Invariant Measures 35
3.1 The Jump Transformation. ....................... 35
3.2 Construction of Invariant Measures. .................. 43
3.3 Solving the Homology Equation. .................... 52
4 Thermodynamical Formalism for Countable State Systems 63
4.1 Entropy .................................. 63
4.2 Pressure .................................. 67
5 Appendix 78
BIBLIOGRAPHY 82

Introduction

Among the dynamical systems, polynomials and rational systems are the symplest.
Starting with the works of L. Bottcher, P. Fatou and G. Julia, they were studied
throughout this century, and very intensively recently. The Julia set J f of a rational

map f consists of points that behave chaotically under iteration:
Jf = C \ {z : EU 3 2 such that Fnlu is normal family }.

There are various ways to investigate the structure and prOperties of these com-
plicated sets in the plane. One of the possible ways to study the dynamically relevant
measures supported on the set. We will be interested in 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.

Let Q = C \ K be an open set on the Riemann sphere C and let w9(E,z) be
the harmonic measure of E C K with respect to (2, evaluated at z E (2. During last
several years there was a considerable interest in the metric properties of such sets.

In particular, the estimations (and even calculations) of
Hdim(w) déf inf {Hdim(E) : E is a Borel support of harmonic measure w}

have been done. Here the symbol Hdim stands for the Hausdorff dimension. As a

result of this attention and especially due to works of Carleson [Ca1]-[Ca3], Makarov
[Mal], Jones, Wolff [JWl], [Jw2], Wolff [W], and Bourgain [B] the structure of the
harmonic measure of general plane sets become much more comprehensible. The deep
analogy between the behavior of sums of (almost) independent random variables
and the behavior of the Green function of the domain plays a crucial part in this
subject ( [Ma2] ). This analogy becomes still more conspicuous if a domain for which
the harmonic measure is investigated has regular self-similar structure. As Carleson
showed in [Ca3] the methods of ergodic theory turn out to be relevant in this case.

It is necessary to point out that harmonic measure in dynamical context appeared
for the first time in Brolin’s paper [Br], where it was established that backward orbits
of a polynomial f are equidistributed (or balanced) with respect to harmonic measure
w of the unbounded component of the Julia set J ( f )

A measure a satisfying
(degree f) - MA) = p(f(A)) if f is injective on A

is called balanced measure. The uniqueness of the balanced measure was established
later by A. Freire, A. Lopez and R. Mafie in [F LM], [Man].

Another way to view the balanced measure is to notice that as n tends to infinity,
preimages f ’"(z) have uniform distribution with respect to it, hence one can construct
the balanced measure as a weak limit of the sums

w — lim —1- 2 6y.

n—>oo n
d yEf‘"z

Here 6,, denotes the Dirac measure supported at y. This process can be viewed as

considering the operator L* acting on measures:

_ 1
_degf

 

d(L*#)(y) dl/(f (31))

and analyzing its iterates. This is M. Lyubich’s construction of balanced measures for
rational functions ([Ly]). He also showed that there exists a unique invariant measure
of maximal entropy; it coincides with the balanced measure and has entropy equal
to log deg f, which is the topological entropy of this dynamical system. Moreover,
the balanced measure has nice ergodic properties: it is Gibbs for totally disconnected
Julia sets and for Cantor repellers, it is mixing for polynomials. Later Brolin’s result
was interpreted as the coincidence of w and the unique measure of maximal entropy
for polynomials.

When we have a dynamical system other than polynomial, the natural question of
comparison of these two measures arises. For rational f it was considered by Lopes in
[L0], where it was proved that if 00 E C\J (f ) is a fixed point of f, then it follows from
m = w that f is a polynomial. We will consider the local setting of the problem when
f is defined only on a neighborhood of an invariant compact set J f. The question
is to characterize the situation when w z m, where ”x” denotes mutual absolute
continuity. It certainly happens when f is conformally equivalent to a polynomial.
The first chapter of the thesis covers the converse problem in the case when f is a
generalized polynomial-like map (GPL).

This is also a question of rigidity: the absolute continuity of the two measures
implies strong information on dynamics.

This problem has been investigated by Lyubich and Volberg in [LV], under the
assumption of the hyperbolicity condition. If (f, U, V) is polynomial like without crit-
ical points, w a: m implies that (f, U, V) is conformally conjugated to a polynomial.

Recall that without the assumption w z m the conjugation with a polynomial can

be done, but in the class of quasiconformal mappings. The proof in [LyV] relied on
the hyperbolicity assumption to derive a Boundary Harnack Principle (BHP) for the
Cantor reppeler, to construct invariant harmonic measure 1/ and then to manipulate
the homology equation:

logd—logJuzuof-u

where JV denotes the Jacobian of V. Without a BHP, the proofs in Chapter 1 involve

the ergodic properties of the two measures.

The second part of the thesis investigates the relationship between the harmonic
measure and the Hausdorff measure on the Julia set of a particular class of maps. It
is shown that the Julia set of a rational f can be a really complicated object. For
polynomials, Manning’s formula [Man] and Brolin’s result give the following estimate

of the Hausdorff dimension of harmonic measure on J ( f ):
logd

Hdim w = , S 1
( ) fJ(f)lOgifidw

 

In fact fym log lf’ldw = logd + Z,- G(c,~), where G is the Green function of C \ J(f)
and c,- are critical points of f escaping to infinity (that is lying in an unbounded
component of the complement of J ( f )). This solves the conjecture of Oksendal [O]
for compacts which are Julia sets of polynomials. The general conjecture was solved
in [JWl], where the estimate

Hdim(w) g 1

has been proved for any compact set K . Note that for J ( f ) with polynomial f such

that there exists at least one escaping critical point one can see exactly as above that

Hdim(w) < 1

The existence of an escaping critical point means precisely that J (f ) is disconnected.
Certainly the situations with connected sets are covered by the famous result of

Makarov which deals with arbitrary continuum K:

Hdim(w) = 1.

Coming back to the case when K is a limit set of a holomorphic dynamical system

f one can suggest two conjectures:

Hdim(w) <1 (0.0.1)

for the harmonic measure on J ( f ) unless J ( f ) is connected. This is not true for ra-
tional functions. However Zdunik proved this conjecture for the so-called generalized
polynomial-like dynamics (GPL) f. See [LV] for the definition of GPL.

Looking at (0.0.1) and having in mind Makarov’s result or/and Jones and Wolff
solution of Oksendal’s conjecture one may conclude that the harmonic measure always
find some ”thin” set of exposed points to concentrate on. This makes plausible the

second conjecture that

Hdim(w) < HdimJ(f) (0.0.2)

for the harmonic measure on J (f ) unless J (f ) is connected. We certainly cannot
expect this to happen for an arbitrary compact set K. This is clear from the example
of Ch. Bishop [Bi] : for any 6 < 1 Bishop constructed a set K such that H dim(w) =
H dim(K ) = 6.

However there are many indications that for K = J (f) the conjecture is correct.
First of all Zdunik proved that the Hausdorff dimension of the maximal measure
is strictly less than the Hausdorff dimension of J ( f ) unless J (f) is connected for

polynomials f. Now Brolin’s result shows that (0.0.2) is true for polynomial dynamics

f. For various types of GPL f (0.0.2) was shown in [MV], [V01], [V02]. Finally let
us mention an interesting result of [Ba] which is in the same vein.
In the last chapters (0.0.2) is proved when f is a parabolic Blaschke product. It

is worthwhile to mention that then the assumption
J (f ) is disconnected

has an ergodic theory interpretation. The fact that J (f) is disconnected means exactly
that f acts non-ergodically on the unit circle T with respect to Lebesgue measure
on the circle. In local terms this means that f has only one petal at the parabolic
points. The reader can find the discussion of these relationships in Aaranson’s papers
[A1] and [A2].

The proof has an analytic part (Chapter 2), dynamical part (Chapters 3, 4) and a
part that mixes analysis and dynamics (Chapter 4). I think that the analytic part is
interesting in its own right. The essence of it is a Boundary Harnack Principle (BHP)
for the Fatou set of f. There is an extensive literature on BHP and the reader may
consult [An], [Wu] or [JK]. It has been recognized in [V02] and [LVo] that BHP may
play an important part in metric estimates of harmonic measure on discontinuous
fractals. But in all these works mentioned above, the existence of BHP relied upon
the fact that the domains under consideration have good geometric localization. They
are NTA, John or Lipschitz domains. The Fatou set of a parabolic Blaschke product
with one petal does not have any nice localization. The complementary intervals of
the Julia set are too small with respect to the distance to the parabolic point (see
Appendix). However, Chapter 2 contains a certain BHP which is one of the key points
in proving (0.0.2). Another key place is Chapter 3 where we use [DU3] extensively.
We couple here our BHP and the technique of [DU3]. After this we prove that

the harmonic measure and the 6-packing measure on J (f) , ((5 déf H dim(J ( f ))) are

singular. Here one comes to an amusing contradiction: if they are not singular then
f can be linearized simultaneously in a common neighborhood of different repelling
periodic points and the parabolic fixed point. So these measures are in fact singular.
But this is a much weaker statement then (0.0.2). To finish the proof we need a
third key consideration which amounts to thermodynamical formalism for certain
countable state systems with potentials that can be unbounded. Our potentials are
of very special kind and this enables us to adapt certain results of [B0] to our case.
This is done in Chapter 3. After that (0.0.2) follows easily.

We are in the position to state (Chapters’ 2, 4) main results. Let D denote the
unit disc. The holomorphic coverings D ——> D of finite degree d are called Blaschke
products of degree d. By simple conjugacy we may consider them as coverings C+ —->
C+ that fix 00. We will freely use this two representations. We consider only Blaschke
products with parabolic fixed point. As a function with positive imaginary part in

0+, a Blaschke product of degree d which fixes infinity can be written as

Ck

f(z):z+CO—Z-:

k=1 Z—flik

 

where Co E R, ck > 0 for k = 1,. . . d — 1. Then f has a petal or two at the parabolic
point p = 00 depending on whether Co at 0 or co = 0. The former case happens if
and only if J( f) i R.

The Main Result Let f be a Blaschke product such that its Julia set J (f ) has
a parabolic point with just one petal ( then the Julia set J (f) is a disconnected subset

of the unit circle ). Let to be the harmonic measure in C \ J ( f ) The following holds:

Hdim(w) < Hdim(J(f)) [I]

The main analytic tools used to construct and compare the harmonic and the

6-conformal measures are the following two results:

Theorem ( Boundary Harnack Principle) If u ,v are positive harmonic func-
tions in Q , vanishing continuously on some uniformly perfect set K contained in R,
and satisfying

u(z) = u(‘z‘) and v(z) = v(§)

then the function log% is Hb'lder continuous of order a on 9U K. C]

Lemma (On harmonic rigidity) Let u, v be two non-negative subharmonic func-
tions in a disc B with diameter I. Let J C I be a closed, uniformly perfect set
with infinitly many components.

If u,v vanish on J and are positive and harmonic in B \ J , and if

 

= lim — = |A(.7:)|2 , Va: 6 J (0.0.3)

for some holomorphic function A in the ball B , then |A| E constant. C]

CHAPTER 1

Generalized Polynomial Like

Systems

If f is a rational function, it was proved by Lopes in [L0] that if 00 E C \ J (f)
is a fixed point of f, then it follows from m 2 no that f is a polynomial. We will
consider the local setting of the problem when f is defined only on a neighborhood
of an invariant compact set J f. The question is to characterize the situation when
w z m, where ”:3” denotes mutual absolute continuity. It certainly happens when f
is conformally equivalent to a polynomial. This chapter covers the converse problem
in the case when f is a generalized polynomial-like map (GPL). As the first section
shows, this problem is related to the Straightening Theorem of Douady and Hubbard
[DH].

1.1 Necessary and Sufficient Conditions for Con-
formal Conjugation

Definition: Let U, U1, U2, . . . Uk be k + 1 topological discs with real analytic bound-
aries such that Ui C U, i = 1,2,...k; UiflUj = (M are j. A map f:Uf=1U,-—+ U

which is a branched covering of degree d,- < 00 on each U,- is called a generalized

9

10

polynomial like system.

Then K, = nnZO U", where U" = f‘"(U). We call J; 2 8K; the Julia set off.
It is also the boundary oono(f) = C'\Kf. The degree off is d = d1+d2 + . . . +dk.
If k = 1,d 2 d1 2 2. we say that f is polynomial~like (in the sense of Douady and
Hubbard [DH]) .

Saying that two maps f,g are (conformally) conjugate means that there is a

(conformal) conjugation in some neighborhoods of the Julia sets.

Theorem 1.1 (Straightening Theorem) Every polynomial like system (f, U, V)
is quasyconformally conjugated to a polynomial of the same degree as f. Moreover, if
K I is connected then the conjugating map is unique up to an afine transformation.
Julia sets of polynomials are uniformly perfect, a property that is preserved by
quasiconformal maps, so the Julia sets of GPL are also uniformly perfect, in particular
regular for Dirichlet’s problem.
If (f, U, V) and (9, U, V) are two polynomial like systems with connected Julia

sets, we say that they satisfy the (BiHolo) condition if
3975 3 U1\Kf —~> U1\Kg biholomorphic, such that d o f = g o d

on the neighborhood U1 of K f In [DH] the external map of a polynomial was con-
structed. For a polynomial like (f, U, V) with a connected Julia set, the external
map h f can be obtained as follows: let a map conformally V \ K , onto some stan-
dard annulus {z,1 < [z] < R} such that 0K; is mapped to the unit circle. Let
W+ :2 a(U \ K I) , let W_ be the image of W+ under the reflection z —+ 1/2 and let
h+ = a o f o a" : W+ —+ {z,1 < [z] < R}. By Schwartz’ relection principle, h+
extends analytically to W+ U W_. The restriction hlsl is an expanding real analytic
map. We will denote it by h f. This construction can be generalized to any polynomial

like system (see [DH]).

11

Definition Two polynomial like systems (f, U, V), (g, U, V) are externally con-

jugated if the following conditon, refered to as (ExtMap) is true:

3d) : S1 —) S1 real analytic, such that (b o h, = hg 0 (b

If (f, U, V) and (g,U, V) are two polynomial like systems with connected Julia

sets then the conditions (BiHolo) and (ExtMap) are equivalent.

Theorem 1.2 Let (f, U, V) be polynomial like of degree d. Then f is holomorphically
conjugated to a ploynomial if and only if f is externally equivalent to z —) zd.

We are going to use the following criterion for conjugation (compare to Shab
and Sullivan [88]): Let h : S1 —> S1 be analytic and expanding. If the measure of
maximal entropy of h is nonsingular with respect to the Lebesque measure on S1 the
h is analytically conjugated to z —> 2".

Since Lebesque measure on S1 is sent by a"1 to the class of the harmonic measure
on J f, the problem of conformal conjugation to a polynomial is reduced to comparing

the maximal measure m f and the harmonic measure wf on J f .

Theorem 1.3 Let (f, U, V) be a polynomial like system of degree d. Then the fol-

lowing are equivalent:

1. 3H a conformal isomorphism in a neighborhood of K, and a polynomial P of
degree (1 such that f = H"1 o P o H.

2. Ed a conformal isomorphism in a neighborhood of K f and a polynomial system

(g, U, V) of degree at such that f = d“ o g o (b and toy = mg,

A GPL system (g, U, V) satisfying Log 2 my will be called maximal. Recall that

by [Br], all polynomials P are maximal; this can also be derived from the following

12

property of Green’s function in C \ K p with pole at infinity: G (P(z)) = G (2) The

following result was proved in [LV] (also see [BPV]):

Theorem 1.4 Let (f, U, V) be GPL. then the following are equivalent:

1. 3d) a conformal isomorphism in a neighborhood of K I and a GPL system
(g, ~ , V) of degree d such that f = (15—1 o g o (t and wg = mg.

2. there exists a function r satisfying :

1) T is subharmonic in U, r 2 0

2 r vanishes oan; r>0 onU\Kf

00

)
) 7' is harmonic in U \ Kf
) T(fZ) = dT(Z)-

Such a function T will be called an automorphic function.

4

Proof : The complete proof of this theorem can be found in [BPV]. Since some
of the arguments in the implication 1 —> 2 help in understanding the construction of
the automorphic function (next section) I decided to include it.

Assume that the system (9, U, V) is maximal; we need to construct an automorphic
function for this system (clearly the existence of automorphic function is conformally
invariant). Let {ax-(E) :2, denote all g-preimages of 5 6 J9 counting with multiplicity.
Let u E C (J9) and g : E ——> g(E) be injective. Then the fact that the Jacobian Jm
equals (1 implies

d—1/9(E)[uo (g | E)"]dm = [Eudm

Since the maximal measure has no atoms, m—almost all J9 can be covered by d disjoint

sets E1 . . . Ed such that, on each E, g is univalent. Thus

(1
/, Zu(x.(£))dm(a = d udm

9 i=1 J9

13

The maximality assumption serves to claim that

d
(J, z.Z::1u(x,-(€))dw(§)= d/Jg udw (1.1.1)

Define <I>(z) = by 9&9, and F(z) = f W —de9 1:1? E Hol(U\Jg). Let us prove

that F E Hol(Ug). To do this choose a contour C in V \ U. Then for every n 2 0

zndz

.f—z

n _ g’(z)z”dz
[CZ F(z)dz _ Jgdw(€) Cm—d Jgdw(€)/C

 

d
= L;(x.(t))"dw(o—d/J sndwe>=0

according to (1.1.1). This proves that the singularities of F are removable. In other

words

‘1’(9(Z))9'(Z) - d‘NZ) = A(Z) E H01(U)-
As Green’s function G satisfies G (2) = f log |z — £|dw(§) + const, we rewrite this line
using the notation G’ for 5‘20 2 (I), and H’ for b'aZH'

G'(9(Z))g'(2) - dG'(Z) = H'(Z),

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

V(Gog)—d-VG=VHandso
Gog—d-G=H+constdéfHo (1.1.2)

Two cases may occur: a) H0 :— 0 in U, h) {z 6 U : H0(z) = 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 Jg, N C U, put F : {z E N : H0(z) = 0}. Then

14

F = U]; I}, where each I“,- is a real analytic arc. Clearly I‘ covers J9 (as H0 restricted
to J, equals (G o g — G), so it is zero). As Jg has no isolated points we can throw
away those I} for which #(Jg 0 Pg) < 00. After this operation the rest of I‘k will
cover J9. So let Jg C F0 2 [:1 F,- and #(J9 (ill) 2 00, i = 1,. . .,m. Now it is clear
that g‘1(I’o) C PD.

We call a cross-point any point of J, which is an intersection of two different arcs
I}, i = 1,. . . ,m. If p0 is a cross-point then the set g‘"(p0) consists of cross-points
(as g‘ll’o C F0). But the number of cross-points is obviously finite. So there is no
cross points at all.

Let (90 be a thin neighborhood of I‘D in which a holomorphic symmetry z —> z*

with respect to F0 is defined. In 01 = 9‘100 we then get

gz“ = (gz)"‘. (1.1.3)

Let us put G(z) = G(z) + G(z*), H0(z) = Ho(z) + H0(z*). Then (1.1.2), (1.1.3)
give us

(Gog — dG)(z) = H0(z), z 6 (9-1.

By definition H0 E 0 on F0. But also this function is symmetric with respect to F0
and so 871:9 E 0 on F0. Thus H0 E 0 on 01 and we have a neighborhood 01 of J9 in
which

Gogzd-G. (1.1.4)

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

the same automorphic property (1.1.4) and the implication 1 —> 2 is proved.

15
1.2 Construction of Invariant Harmonic Measure
and of the Automorphic Function

In order to construct the automorphic harmonic function we need an invariant version

of the harmonic measure.

Theorem 1.5 Let (f, U, V) be a GPL. Then there exists a finite measure V on Jf
such that V is f -invariant and u z w,
Proof: We will use the following result of Y. N. Dowker and A. Calderon which

can be found in [F0] (reformulated in a convenient form):

Theorem 1.6 Let u be a probability measure on a compact set X. Let
T : X —+ X be a continuous endomorphism such that u is completely non-singular
with respect to T. Then there exists a T-invariant probability measure A absolutely

continuous with respect to u if and only if the following holds
u(E)<1=> supu(f—"E) <1

If ,u is ergodic then )1 is ergodic.

Proof: Fix an arbirary Borel set E C J, w(E) = 1 — e < 1. Let I‘ be a smooth
curve encircling J and separating it from 8U and let I‘” = f “”(I‘). The main things
now are six notations.

Let w, v denote the harmonic measures of E with respect to Aoo( f ),
U \ K 1‘ respectively. Let V,W denote the harmonic measures of J \ E with
respect to A00, U \ K f respectively. For any function (15 let d" denote d o f" where
defined. As usual (129(5, z) denotes the harmonic measure of S evaluated at z with
respect to 52.

First we need a simple lemma. Fix a compact set K in Q and consider two

16

harmonic measures on an - with respect to S2 and with respect to Q \ K evaluated

at the same point a E Q \ K.

Lemma 1.7 The two harmonic measures on 89 are boundedly equivalent.

Proof: We present the proof in the case when all points of 60 are regular. Only
this case is used in what follows. Let 0 be a neighborhood of BIZ with smooth
boundary and satisfying 0 fl(K U{a}) = 0. Let G, g be Green’s functions of Q, Q\K
respectively, with pole at a. Then by Harnack’s principle CG 3 g _<_ G on the
boundary of 0. By maximum principle this inequality extends to 0 (both functions
vanish on 69). Then clearly the functions g — cG and G — g are subharmonic in O
and so their Riesz measures are nonnegative. The Riesz measures of G and g being

equal to our harmonic measures, we are done.

Coming back to the proof of Theorem 1.5: we must show that

ono(E,oo) S 1 — 6 => ono(f_"E,oo) g 1 — 6.

As w(oo)$1— e, we get W(oo) 2 6.

Then W({) 2 cc on I‘ and by previous Lemma, V({) 2 6 on I‘. Then V"(§) 2 (5
on P". But these functions vanish on 8U" and so ono(f‘"(J \ E),€) Z 6 on 1‘". So
ono(f’"(E),{) _<_ 1 — 6 on 1‘”. As I‘n separates J from 00 we obtain w(f‘"E) =
ono(f""(E),oo) g 1 — 6.

We are going to prove that the just constructed invariant harmonic measure 1/ is

boundedly equivalent to (1).

Theorem 1.8 There exist constants 0 < c1, 02 < 00 such that

17

Proof: Let us prove first the right inequality. We wish to repeat the above
considerations but it seems hard to get rid of the influence of EU. However we are
going to prove that w(E) _<_ 6 => w( f ‘"E ) s cc for a certain finite c. The construction

of invariant measure in [F0] then gives the right inequality. Let us fix I‘ as above and

such that
1
< -.
rpearxmeAaUé) _ 2
Then if (2,, is any component of U" and 7,, = Fnflfln, we can write

wU\K;(3Ua f"(£)) = wnn\K,(3Qm€) and thus

. € 6 7.. (1.2.1)

[\DII---I

an\Kf (Ban: 6) S
We start with the chain of implications:
w(oo)§c :>w(£) ngc forfiel‘ =>

=> v(€) S Cpe for 5 6 I‘ => v"(€) S Cpe for g E 'y,,.

Let us compare u1(€) = v"(§), with u2(§) = 01A... (f ”‘E , f) on 7,, for each component

S2,, of U ". By Poisson formula in (2,, \ K f we have:

112(5) — (a... U2(n)dwn.\x,(n.€) = we

Let “2(50) z maxvn maxtevn “2(5) = maxter" ”2(5)-

Then By (1.2.1) we have u2(§0)(1—%) g v.1(fo) g Cpe. But I‘" separates J; from
00 and so

ono(f”"E,oo) = u2(oo) g 2C1~e

The left inequality can be proved exactly the same way.

18

Theorem 1.9 Let (f, U, V) be a GPL. Then the following assertions are equivalent:
1. f is conformally maximal
2. wf z mf.

Proof: We will be using the following notations: if u is a subharmonic function
then pa 2 Au is its Riesz measure. Let G be Green’s function of A00( f) with pole
at 00. We know that pa = w. Let (I) = 2&1, 45 :- log (I). Clearly (I) is bounded
away from zero and infinity (this is just Harnack’s inequality essentially). It will be
important for us that (I) is the Jacobian of to with respect to f. Let p = fi, 7 = log p.

The measures V, m are finite, invariant and ergodic. So u z m implies z/ = m. We

start with the homology equation:

d—logdz'yof—y, w—a.eonJ (1.2.2)

This is obvious from the computation of the Jacobians of the measures 1/ = m.

To prove our result it is sufficient (and necessary) to construct the automorphic
harmonic function r.

The first step is to find a disc B = B(x, r) centered at the Julia set and to

construct a nonnegative subharmonic function u in B such that

do)
dun

 

= e7 on B (1.2.3)

The function T will be an extension of u if the Julia set J I does not lie on an ana-
lytic curve. Otherwise take 7' to be an extension of the average between 11. and the
symmetrization of u over the analytic curve.

Let F be be a set with w(F) > O on which 7 is continuous. Let F0 C F, w(F0) > 0

19

be such that

 

limw(B(y.6)flF)
H0 w(B(y.6))

= 1 (1.2.4)
Let (J, f, 771) be the natural extension of (J, f, m) to the space of inverse orbits of f,
with f being the left shift. We denote by it : J —-> J the projection onto the ”0”
coordinate. Then m(7r‘1(F0)) > 0 and by the ergodicity of fn one can choose 5: such
that f‘"(x) E 7r‘1(F0) with positive frequency. In particular we have chosen :1: E J f
and a sequence of compatible inverse images x,, of x such that x,, 6 F0 with positive

frequency. But one can do more (see [F LM], [Z1]): we can choose B = B(x, r) such

that on 38 there are univalent compatible inverse branches F,, such that
diamF,,(2B) g e-"5

x,, = F,,(x) meets F0 with positive frequency (1.2.5)
Let us consider the family '11,, = d"G o F,, in BB. Then by (1.2.2)

dw

d” (y) = 67(y)—7(Fn(y)), y E Jf n33.

 

In particular ” ,uun Ms C < 00 and moreover clw g #11,, _<_ cgw (see Theorem 4 which
gives the boundedness of 7).
Let {nk} be a subsequence such that 7(x,,,) —> c, k —> 00, $11,, 6 F0. Without

loss of generality we can think that

i7($nk) — C] S 2-1::

w(Fnk(2B)\F) —k
w<F..(2B>) 5 2

 

20

The last assertion follows from (1.2.4), (1.2.5). Let E), = F,;1(FflF,,k(2B)). Then

 

[did - BAH] S CIZ—k 011 El: (1.2.6)
"I:
,1,,(2B\E,,) g 022-k (1.2.7)

Let K be a relatively closed subset of the disc B. We denote by S+(B, K) the set of
bounded subharmonic functions in B vanishing on K and positive and harmonic in

B \ K. We use two results from potential theory:

Lemma 1.10 Let {v,-} be a sequence of uniformly bounded functions from S+(B, K).
Let K be regular for the Dirichlet problem in C \ K. Then there exists a subsequence

which converges pointwisely to a function from S+(B, K).

Lemma 1.11 Let u belong to S+(2B, K), for some ball 28 of diameter less than 1
and having cap(BflK) > 0. Then

51ng S 03.x II a. II -

So as I] punk Ms C we conclude that u", are uniformly bounded.
We may think that the subsequence in Lemma 1.10 is {unk} itself; put
u0 = lim;HOG u,,,c in B. The convergence is pointwise bounded and so Hun, ——+ um,
weakly. But (1.2.6) and (1.2.7) show that punk —> e"7+cdw weakly. Thus
dam, = e’7+cdw and uoec satisfies (1.2.3).

Now the construction of 7 follows word by word the construction in [LV] and [BV].
For the sake of completness, here is a sketch: Consider By a component of f ‘"B and

define r on it as follows:

I *-*

7(2) “‘é’

nU(f"Z)

9..

21

In [LV] or [BV] it is shown in detail that r (or its symmetrization over J i, if J f lies
on an analytic curve) does not depend on 6 or n, that is if B9, 0 B92 ¢ 0 then r (or
its symmetrization) is the same on this intersection. This follows quite easily from
the homology equation. Now 7' is defined on the set 0 = U f ”B.

To define r on U use the homology equation to push forward the extension: if
7(x) has been defined, r(f"(x)) dfif d"r(x) . Since the backward orbit of any point
in U is dense, eventually one of its preimages lands in B, where T has been defined.
This extension is well defined in U, harmonic and satisfies the homology equation on

an open subset of each component of U, therefore in the whole U.

This concludes our construction and proves the theorem.

CHAPTER 2

Boundary Harnack Principle

2.1 The Main Lemma.

Definition A compact set K is uniformly perfect if

cap(KfflB(x,r)) Z c cap(B(x,r)) Va: E K; and ‘v’r S r0

Through this section we are going to work with domains SI of the form 9 dg
B(0, R) \ K for some uniformly perfect compact set K C R and for some ball
3(0, R) with dist(K, 68(0, R)) 2 diamK .

We will denote by (2+ (1;! Qfl{z : 3z 2 O} and by (2. (1g Qfl{z : Sz 3 0}. Given a

point 5 E K we will denote by 5’+ déf 5 + ir/2 and by 5"“ déf 5 — ir/2.

We are going to deduce a couple of Harnack-type results in 9+. Some of them
are inspired from [JK] . Most of the constants M that appear depend only on the
constant c from the definition of uniformly perfect sets ; when this is the case we

will write M = M (unif.perf)
Proposition 2.1 There exists a universal constant M1 and some positive 60 ,

22

23

such that for any 6 _<_ co and any two points 21,22 6 9+ satisfying dist(z1,22) <
2’95 and dist(z,-,BQ+) > 6 can be joined by a chain of at most Mlk balls

B(c1,r1), . . . , B(ckI, rkr) contained in Q and having

1
r,, 2 fidist(z,-,B(cp,rp)) and rp 2 dist(B(cp,rp),6§l) Cl
1

Proposition 2.2 There exists some universal constant M2 such that for any func-
tion u positive and harmonic in Q and for any z1,22 6 9+ satisfying dist(zl, 22) <

2% and dist(z,-, 89) > e , the following holds:

u(z,)

T; S ’U.(ZQ) S M; “(21) Cl

Proof. Use Harnack’s principle k times.

Remark For any M1 there exists M2 such that if

§

 

(21) _>_ M; and dist(z1, 22) < Mfe (2.1.1)
u(z2)

for some 21,22 6 {2+ then min,=1,2 dist(z,~, 852) < 6

Lemma 2.3 There exists M3 = M3(unif.perf) such that for all qo E K and for
any u positive harmonic function in {2 such that u vanishes continuously on

B(q, r) (I K the following holds:

sup{u(x); x E B(q0,s/M3) } g sup{u(x) ;x E B(q0,s)} for all s < r El

1
2

24

Proof: Denote by a = sup{u(x), x E B(qo, 3)}. Then
“(33) S a wB(qo,s)\K(aB(QO13)1$)
By the uniformly perfectness of K there exists some 7 = 7(unif.perf) such that
w3(qo,s)\K(dB(QO, 8),x) S 7 < 1 for all x E B(CIo, 3/2).
Let M3 = 2" for some n large such that 7" _<_ %. Let z E B(q, S/M3). Then
u(z) s 7% s gsupne), 2: 6 Ba... 5)} 1:1

Lemma 2.4 There exists M4 = .M4(unif.perf) such that if qo E K and ifu is a
positive harmonic function in 9 that vanishes continuously on K flB(qo,2r) then

the following holds:

u(x) 3 M4 [u(q6+) + u(qg’)] for all x E B(CIo, r) E]

Proof. Let x 6 (2+ . Let M3 as in Lemma 2.4. Let M2 satisfy:

6

u(zl)
< M;

”(22)

 

> M; and dist(z1,22) < e :> min dist(z,—,K)

Let N be large such that 2” > M2. Claim that u(x) g M2N+4 u(q5+) for all
x E B(q0,r)ߤ2+.
If not, 3331 E B(q0,r) such that u(xl) Z M2N+4 u(q6+). By

’U.(.’L'1)

“((16+)

 

Z Mr?“ and dist(x1,q6+) < 2r

25

we have dist(x1, K) S W' Let q, be a point in K closest to 13,. Then dist(q,, q0)<

r + film. By Lemma 2.4

SUp{U($);x 6 BMW/Mil} 2 2” sup{U($);$ E B(q1,r/Miv+3)}

2 2Nu(xl) Z Algu(x1) > Mil/+5 u( q6+) (2.1.2)

In this manner we can find 332 E B(q,,r/M33) such that u(x2)_ > M2N+5u (q6+)

By (2.1.1), dist(x2, K) S 17:53. Let ([2 be a point in K closest to x2. Then

dist(q2,q0) S r + 4r/M;[V+4 + r/ll/Ié’ + 4r/ll/Iév+5 S Zr

and
sup{u(x), x E B(q2,r/M§’} 2 2” sup{u(x), x E B(q2,r/M§V+4)}
2 2Nu(x2) > M2N+6u ( (15+) (2.1.3)
Finally we can find a sequence of points x1,x2,... such that x,, —) K with

dist(x,,,q0)< 2r and u(xn) 2 A42’V+”u(q5+) ——> oo . This is impossible because u

vanishes continuously on B (qo, 2r). Therefore the claim is true.

Theorem 2.5 If u ,v are positive harmonic functions in Q , vanishing continuously

on K, and satisfying
u(z) = u(E) and v(z) = v(E)

then the function logg can be extended to a Holder continuous function of order
a = a(unif.perj) on QUK. E]
Proof. Let q E K and r > 0 be fixed. Denote by F,, (1;! BB(q,r/4") and by

 

F'“?

26

Pn+1/2 (1;; BB (q, r / 4”+1/2). Following [LV] it is enough to show that

, u u maxn 9
1=m1n— S max— S1+€,, => ——ri37” S 1+,86,, (2.1.4)
F.. v Fn v mmpn+1 5

for some [3 = B(unifperf) < 1.

Decompose I‘,, into

I‘L={1+%’<%S1+en} (2.1.5)
and
1‘3 = {1 g 3 g 1 + 523} (2.1.6)
'0

We are going to use the following harmonic measures:
w(-) = wot, 00)

wl‘n(°a 6) : wB(q,r/4")\K(°a g) for S E B(q, T/4n)

wn+% ('7 S) : wB(q,r/4"+1/2)\K(’v S) for S E B(q, T/4n+1/2)

By Poisson’s formula we get:

M6) = /, mama + [1, u(mdwnm)

n

Denote by ’UL(€) = frg, v(fl)dwr..(77,€) and by “05(5) = frg v(nldwr.(n,€)
Then v5 + (1+ c,,/2) vL S u(5) S (1 + e,,/2) v5 + (1 + c,,) vL

LGt §+ E Fn+1/2 be 6+ = (Ii-1774M”2 and 5- E Fn+1/2 be 5- = q — ir/4”+1/2

Suppose that

US(€+) Z UL(€+) (2-1-7)

Then the same is true for 5-.

27

Claim: There exists c = c(unif.perf) such that

vS(5) 2 c vL(5) for all5 E Pu“. (2.1.8)

Assume the claim is true. Then by (2.1.5) and by vL + v3 = v we get

“(6) s (Hen/2) 115(5) +(1+e..) 1H0 _<_ (1+en/2)v(£) + en/vac)

 

 

therefore
1
21(6) 5 (1+ en/2) 22(5) + e../2 1 + C v(é) 3 (1+ a.) 22(6)
and
———mfxr"+’§ S 1 .+ fie: = (1 + hen)
mlnpn+1 3 mmpn 5

If the opposite of (2.1.7) is true, one can prove that the Opposite of the inequality

(2.1.8) holds and the minimum of 35 can be estimated from below in order to deduce

(2.1.4).
Proof of the claim: Let 5 E Tn“. For convenience suppose that q = 0. Decompose

Fn+1/2 into four arcs

 

 

. . 1' . . r
_ arr/20 119n/20 _ _ :191r/20 i211r/2O
I+_(€ ,6 )4n+1/27J —(€ )8 )4n-I-1/2’
. . 7‘ . . 7'
_ _ i217r/20 139n/20 _ z391r/20 21r/20
I _ (8 ’8 )4n-H/2’ + _ (e ,e )4n+1/2'

The arcs 1+ and I _ are far from the boundary of (2. By Harnack’s principle v(n)

and v(5+) are comparable for 77 E 1+. Therefore

v5(n)dwn+g(n.€) 2 C1 v3(€+) wn+,(1+ UI—.€) (2-1-9)

3
v (f) 2 /I+UI_

28

and

UL(€) S C2 vL(€+) wn+%(I+ UI-vE) + vL(7l)dwn+%(Tl1€) (2H110)

J+UJ_
The estimate for vL will be done in two steps:
1. Prove that vL(5) S c’ vS(5+) for 5 E J+UJ_.
2. Prove that wn+%(J+UJ_,5) S c’ wn+%(l+ UI_,5+) for 5 E I‘,,+1.

Once the two steps are concluded the proof of the claim becomes trivial.

Step 1. Consider the smallest disc that contains J+. If it doesn’t meet K there is
nothing to prove. If it contains a point go of K, apply Lemma 2.6 for the function

vL and the disc B(qo, 2diam(J+)). We get that
v"(€) S M4l vL(q6+) + vL(€I6‘)l S Mil v50?) + "03((16') ]. V6 6 J+

By the Harnack principle vS(q6+) and v5(5+) are comparable.

Step 2. Let Gn+% be the Green function in B(q,r/4"+1/2) \ K. Denote by c] the

middle point of J+; let (1) be a smooth function such that
d = 1 on B(cJ, |J+|) and d) = 0 outside B(CJ,2|J+])
We get:

.J, </ AG 1 , d s
wn+§( + g) — B(CJ,]J+]) n+2(z 6) Z

< ¢(Z)AG,,+%(Z, 5)dz =

_ B(CJ.2|J+|)
= / A¢<z)c..1<z,odzo (2.1.11)
B(CJ,2|J+|) 2

Recall that 5 is a point in I‘,,+1 . Apply Lemma 2.6 to the harmonic function

29

Gn+%(-,5) and the ball B(CJ,2|J+|) in the domain B(CJ,10|J+]). To simplify nota-

tions letp+=cJ+iJi2t1 andp_=cJ—ilJ—;l .We get:

ammo s M. [Gn+%(P+,€) + Gagpsoi for all z e B(c.J,2IJ+I) (2.1.12)

 

Use (2.1.13) and the fact that the function <15 has IAqS] S C835} in the inequality

(2.1.11). We get

wn+%(']+1€) S M5 [Gn+%(p+’€) + Gn+%(p-a€)l (2'113)

Comparing Gn+%(-,p+) and wn+%(I+ UI_, ) on the domain

B((Io, W) \B(p+,100—4:m§) \K we get that
Gn+%(5,p+) S const wn+%(1+ UI_,5) for all 5 E I‘,,+1
The last inequality and (2.1.13) conclude the proof of Step 2. [3

Corollary 2.6 If J I C R is the Julia set of a rational map and if G is the Green
function in C \ J I, then the function log egg) can be extended to J f as a Holder

continuous function.

2.2 Harmonic Rigidity.

Consider u,v two positive, harmonic functions in a domain It, vanishing on 69.
that extend as nonnegative subharmonic functions in the plane. If the boundary of
the domain is ”nice”, one can prove that the ratio u/v has Holder extension to the

boundary. This limit relates to the Riesz measures of the subharmonic functions u, v.

Lemma 2.7 (Grishin’s Lemma) Let Q be a neighborhood of a compact K and let

30

u, v be two subharmonic functions in Q, vanishing on K and positive and harmonic

in Q \ K. Suppose that for every x 6 UK there exists the limit

lim u(x)
5—)x,5€V\K v(x)

 

120?) =

and this limit is continuous. Then dun = p duv, where pa, [1,, are the Riesz measures
of u and v.

See [Gr] and [F] for proof.

Lemma 2.8 (Volberg’s Lemma on Harmonic Rigidity) Let u,v be two non-
negative subharmonic functions in the disc B with diameter I. Let J C I be a
closed, uniformly perfect set with infinitely many components and with linear measure
H1(J) = 0.

If u, v vanish on J and are positive and harmonic in B \ J , and if

 

u(x) : lim u_(z_) = |A(x)|2 , Va: 6 J (2-2-1)

for some holomorphic function A in the ball B , then IA] E constant.
Proof: Let us consider Bu(z) and 6v(z). These are holomorphic functions in
B \ J and their 5 derivatives are the Riesz measures nu, av, respectively. Consider

the following function:

 

w1(z) = 6u(z) — A(z)A(Z)8v(z).

It is holomorphic in B \ J and its distributional 3 derivative is equal to

5w, 2 uu— |A(x)|2u,, E 0

by the assumption (2.2.1) . So w1(z) is a holomorphic function in B. In a similar

31

way one can show that wg (1g % — A(z)A(Z)g—:.— is anti—holomorphic in the ball B.

In particular 6u(x) — |A(x)|26v(x) is continuous and locally bounded on I ,
the diameter of the ball B.

Without loss of generality we can assume that u(5) -.: u(z), v(z) = v(z) ;
otherwise we can consider u*(z) = u(z) + u(z), v*(z) = v(z) + v(Z) and (2.2.1) still

holds with symmetrized functions.

But if u, v are symmetric, then on I we have Bu 2 %, 8v = -g—: and we

conclude that
Bu

— — (x)]2——’i is continuous on I (2.2.2)

83: 83:

Next we will prove that W(x) déf u(x) — |A(x)|2v(x) is locally G1 on I.

Byg’5=(%+a%) wegetthatfora,bER ande>0

b6W

 

 

 

W(b + is) — W(a + is) z/a 57“ + is) + 2323“ + ie)dt (2.2.3)
By W(z) = u(z) — A(z)A(Z)v(z) we get that
8W Bu , _ —I— _ _ @
32— - 5 — A ( )A(z)v<z) — A<2>A (z)v(z) A<z>A<z> a,
and
BW Bu _ 0v
5 “‘ g — A(z)A(z)5—Z

This implies that for z E B \ J

96% = g; — mam—61’] — v(zn A’(2)A(i) + Acme) 1+

 

%[WA(Z) — A(Z)A(Z)] = 101 — 1113 'i' 102 + w.,

 

+13; — A(z)7i(—zi% I +

The functions w,, 102, w3 are continuous on a neighborhood of I, because they are

32

either analytic or anti-analytic or the product between two continuous functions.

 

The function w., = g§[A(z)A(z) — A(z)A(z)] is bounded uniformly in 6 because

 

 

 

|A(z)A(z) — A(Z)A(z)] S cc for Im z = e
and
[2— :IS dflv(C)SE forlmzze
JIC — 2| 6

Equation (2.2.3) becomes:
b
|W(b + ie) — W(a + i6)| 5/ |w1(t + if) — w3(t + is) + w2(t + is) + w4(t + ie)|dt

Sending e to zero one gets that |W(b) — W(a)| S M |b -— a], therefore W is absolutely
continuous on R and
BW

E:w1($)—w3+w2+0

is correct for x E I \ J, and therefore on I since H1(J) = 0. Therefore W is in
Clloc(1)

Its derivative is equal to

13—: M)I’g—:l — v( )IIAIAIW

Consider the function W on one interval L of the complement of J in I.

It vanishes at the endpoints of L , so its derivative has a zero at a certain x; E L:
(u —- |A|2v )'(a;,) = 0 (2.2.4)

Because the function u — |A|2v vanishes on J which is a perfect set we can conclude

that its derivative vanishes on J also.

33
On the other hand on the real line

an

(H - IAIQ’UY = B—x

3v , ,
— IAI’-(.,— — v(IAl’) = wl — (IAI’) v
x
and both (u — |A|2v)’ and ([Alz)’v vanish on J , therefore w1(x) = 3% — [AP—3%
has to be zero on J . But this function is holomorphic in a ball containing J, so it

has to vanish everywhere : w1(x) E 0 on I and

g: — [Alzg—Z E 0 on I. (2.2.5)

Computing the the x -derivative in (2.2.4) we get that

03

2 , at) 2 I _
6,01) - lAl Imam) - v(rrz)(|A| )(xz) — 0

which according to (2.2.5) gives that v(x))(|A|2)’(x1) = 0 . The function v being
harmonic outside of J , can not vanish at x), which means that (|AI2)’ (x1) = 0 .

This is enough to conclude that IA]2 = constant .

Corollary 2.9 If one adds u,v are symmetric to the assumptions above, then u =

const v.

Proof: By the previous lemma, A(u — cv) = 0. So the function u -— cv is
harmonic in the ball B. Its zero set, Z (u — cv), is locally a real analytic curve,
otherwise it E cv.

Thus if u 74 c v then at least u = c v on the union of some intervals of the real
line, which means that 3% — c g—Z = O on some intervals on the real line. Due to the
symmetry of u, v the y-derivative of both functions vanishes on R, so the following

holds:

an — c 8v 2 0 on some intervals of the real line

 

34

therefore Bu —— c 0v = O everywhere in B, because it is holomorphic.
Similarly 5n — c 3v 2 0 everywhere in B. This implies that u -— c v is constant
in B , and being zero on the real line, it has to be identically zero in B. Therefore

HEC’U

CHAPTER 3

Construction of Invariant Measures

3.1 The Jump Transformation.

In order to compare the harmonic measure and the Hausdorff measure we would like
to construct some f—invariant measures equivalent to each of them. If we attempt to

construct a finite f-invariant measure equivalent to to, a necessary condition is :
ifaset A C J has w(A) <1 then w(f_"A) <1—e < 1.

By w(f‘"A,x) = w(A, f"(x)) —+ 1 as n —) 00, there is no finite f-invariant measure
equivalent to w.

Given a parabolic Blaschke product f with one petal and the harmonic measure
or on J we will construct a transformation T on an Open, dense subset J, of the
Julia set, called the jump transformation, such that the triplet (w,T, J,,,) satisfies
Walters’ conditions [Wa].

First of all I.) is going to be totally non-singular:

w(E) : O <==> w(TE) = 0 (3.1.1)

35

36

and the transformation will satisfy:

360 > 0 such that Vx E J,, T"’(Bgfo(x) flJ,) 2 LI, A,(x), (3.1.2)

that is the disjoint union of open sets such that T : A, —) 82,0 (x) H J, is a homeo-
morphism and dist(Ty,Ty’) 2 dist(y,y’) \7’y,y’ e A,(x).
Also

V6 > 0 3M such that Vx E J,, T’M is e-dense in J, (3.1.3)

Notice that (3.1.1) implies the fact that the push-forward measure
T*w = w o T‘1 is absolutely continuous with respect to w. Also the Jacobian of

or with respect to T is well defined and we will introduce the notation

clw 0T
dw

 

d) = -10g (:5) = -10sJJ(-"II)

To write down the rest of Walters’ conditions we need more notations:
let SnIp(x) = 2:01 Ip(T‘x) for any function p : J, —> R. Given b an element
of T‘"bo for some b0 6 J,, let T;" denote the inverse branch that sends b0 —> b.
Set maxi = 15mg) — 5.20% for y = Tb-"w = T.-"x'.

Let f} be the class of functions 90 such that

(i) Egg—1,, em’) S Kw < 00 (3.1.4)
(ii) c¢(x,x’) = sup"21 sup c¢,b(x,x’) S CV, < 00 for x,x’ close enough (3.1.5)

(iii) c¢(x,x’) —> 0 as d(x,x') —> 0 (3.1.6)

Remark If Ip is the J acobian of the harmonic measure with respect to the jump

transformation, then

(3.1.4) (1) the density of w o T"1 with respect to w is bounded

37

(3.1.5) 4:) supn sup{%;y 6 T‘"x,y’ E T‘”x’} S A(x, x’) S K < 00
(3.1.6) 4:) A(x,x’) —> 1 when d(x,x’) —-> 0

Let U C (X ) denote the space of uniformly continuous functions on some set X.

For Ip 6 fr define the operator [2,, on UC(X) :

5.0903) = Z e”(’”g(y)

yET—lx

Recall that we have denoted by 6 the Hausdorff dimension of J. We are going
to show that both potentials w = —logJ$ and (b = —6log|T’| satisfy Walters’

conditions (3.1.4) to (3.1.6) so that the following theorem from [Wa] applies:

Theorem 3.1 Suppose T: X0 -—> X0 satisfies (3.1.2) and (3.1.3) and that 90 6 Fr .
Then Ly, extends to a bounded linear operator in C(Xo) .

Furthermore, there exists a unique probability measure u satisfying (3.1.1) and
some A > 0 such that

Egu = /\ 11. (3.1.7)

There exists a unique positive function h, h E C(Xo) such that
£,ph = /\h and u(h) = 1. (3.1.8)

The measure hp is T-invariant.

See [Wa] for details.

This could be enough in order to construct invariant measures equivalent to the
6-conformal measure and equivalent to the harmonic measure. However, we need
better properties of the densities h in equation (3.1.8). For this reason the ac-
tual construction of the invariant measures will be done in the next section, using a

theorem of Ionescu Tulcea.

38

Following M.Denker and M.Urbanski, [DUl], [DU2], [DU3], we will construct the
Jump transformation T for conformal parabolic systems. For the sake of completeness
we are obliged to repeat parts of these works.

Let J be a compact set consisting of at least two points, let J C U C C be an
open neighborhood of J and let f : U —-> C be a holomorphic map. We assume

that there exists a point p such that f(p) = p and f’(p) = 1.

Definition A system (f, U, J) is called conformally parabolic if there exists a

unique parabolic point p in U and if
(i) f (J ) = J
(ii) there are no critical points in J

(iii) there exists a neighborhood W of p such that if x E U satisfies f"(x) E

W forVn>0 thenx=p
(iv) for x E U \ J, 37', > 0 such that f"(B(x,r,,)) —+ p uniformly
(v) if f"(x) E U, Vn > 0 then x E J or f"(x) —> p
(vi) the mapping f : J —> J is topologically exact.

Lemma 3.2 Let V be an open neighborhood of the parabolic point p. There exists
60 = 60(V) > 0 such that for every 2 E J \ V all inverse branches of f" are well
defined on B(z, 360).

Proof: See [DU3].

Let now f be a parabolic Blaschke product with Cantor like Julia set, having
the parabolic point at p = 1 . Denote by I‘“ the cone domain with vertex at the

parabolic point, that is I‘" = convh(1, B(O, sin 01)).

Lemma 3.3 There exists a such for every z E J all inverse branches of f" are

well defined in B(z, r) if B(z,r) 01“" = 0

39

Lemma 3.4 Such a transformation f is positively expansive, i.e.

3 (5' > 0 such that if supd(f”(x),f"(y)) S 6' then x = y.

n>0

Proof: See [DUl].
Recall that a cover 7?, = {R1, R2, . . . Rs}, 3 S 00 of X is said to be a Markov

partition if it satisfies:

(i) R.- = as

(ii) intR, are disjoint

(iii) f (R,) is the union of some Rj.

Assume that the map F : X —+ X is open, surjective and positively expansive. From

the proofs in Ruelle’s book [Ru] the following result can be deduced:

Theorem 3.5 If F : X —+ X is an open, surjective, positively expansive continuous
mapping of a compact metric space X, and if ,u is an atomless measure,
then there exists a finite Markov partition of arbitrary small diameters satisfying:
u(BRl UBRg U. . .8R,) 2 0.

The transition matrix A = (A,,j),,j53 associated with the Markov partition R0

is defined by

Am“ 2

A sequence k1,...,k,, is said to be A-admissible if Akiki+1 = 1 for every

i = 1,. . .n —— 1. Coming back to our parabolic system, given a Markov partition

and some A-admissible sequence k1, . . .,k,, define A(k1,. . .,k,,) = 39:, f"j+1Rkj .

We will call such a set an n-cylinder. The family of all n-cylinders will be denoted R3.

40

Among all cylinders of R3 we distinguish ”good” and ”bad” cylinders. A cylinder
will be called good if f"‘1A(k1,...k,,)flW 2: Q) , and will be called bad cylinder if
fn’1A(k1,.. .k,,)flW 75 (b. The set W is the neighborhood of the parabolic point
from the condition (iii) in the definition of ”conformally parabolic systems”. Let 9,,
be the collection of the good cylinders of order n, let 8,, be the collection of the
bad cylinders of order n, and Rg be the collection of all good cylinders, R3 be the

collection of all bad cylinders.

Remark Since the map f is expansive and so expanding with respect to a
metric compatible with the topology on J, (see theorem 2.2 of [DU2]), the diameters
of elements of R3 tend to zero and so the family R0 generates the Borel o-algebra

mod 1/ for any non-atomic measure V on J.

Lemma 3.6 Let 1/ be a non-atomic measure on J. Then X0 (1g UAERG A satisfies
l/(Xo) = 1 and
"1:11.10 ZAegnl/(A) = 0 (319)

See [DU3] for proof.
Now we are ready to introduce the jump transformation of f.

For every x 6 X0 take the smallest rank of a good n-cylinder that contains x, i.e.
N(x) = inf {n ;x E A(k1,. . .k,,) and f""’A(k1,...k,,)fl W = 0}.

In view of previous Lemma, N (x) is almost everywhere finite with respect to any

non-atomic measure. Schweiger’s jump transformation is defined by
T($) = WWII?)-

The set N “1(n) consists exactly of good cylinders A(k1,. . .k,,) E 9,, that have

41

a ”bad” parent: A(k1,. . .k,,_1) E 8,,_1.
Let us consider a new family, R,, to be the union for all n of families of cylinders

forming N ‘1(n). Denote the elements of R, by 0,. Then (3.1.9) becomes

U C :X0 and T(C') = U 0' (3.1.10)
CER. C’ER.,C’ flT(C)¢0
In this sense R, constitutes a countable Markov partition for T. Also the transition
matrix for (T, R,) is aperiodic.

For x 6 X0 define the positive integers N,,(x) by:
N1(33) : N(x): Nn+1($) : Nn(x) '1' N(Tn($))

which gives T"(x) = TN"(I)(x). If we denote by C"(x) the cylinder of R2 containing
x , then also

71 _ Nut”)
Tlcnm — fl

0%)
The following result is proved in [DU3]:
Lemma 3.7 The jump transformation is expanding, i.e. there exist 6, fl > 0 such

that

I(T")'(x)| 2 c ef’" for all x E X0,n > 0. (3.1.11)

Lemma 3.8 There exist constants B, 17 depending only on f, such that for all pairs

of points x, x’ sufficiently close, if y = Tb-nx, y' = Tb—"x’, then Vn

Tn! I ’
i( )0!” Se—S-dtcx). (3112)

42

Let 2, C Riv be the space of all admissible sequences of cylinders according to
the transition matrix of (T, R,).
Let 22 = {(Cn),,20 E 2, such that 0,, T‘”(C,,) # 0}. The spaces 2, and 22 may
not be compact. Since by lemma 3.8 the diameters of of the partition Rf converge

to zero, one can define the projection 7r : 22 —> X0 setting

Consider the metric p on ES to be
p((Cn)nZOI (C;)n,20) : e— min{n; Cn¢C7’1}
Proposition 3.9 The map 7r : 22 ——> X0 is a Holder continuous surjection, with

exponent 3 from (3.1.11)

Remark If we identify x 6 X0 with r‘lx then one can transport the metric p

to X0. The previous lemma can be formulated as
\7’x,x' 6 X0 |x — x’] = d(x, x’) S cp(x,x')f’

Using the fact that for any cylinder C E R, there exists a cylinder A E R0 such
that

T(C) = f(A), (3.1.13)
one can prove the following lemma:

Lemma 3.10 1. For any 6 > 0 there exists some M such that Vx 6 X0, T‘Mx
is e-dense in (X0,p)
2. For any 6 > 0 there exists some M such that Vx 6 X0, T‘Mx is

c—dense in (X0, d).

43
Using the fact that f is expanding we can conclude that

Lemma 3.11 For every C"1+1 E R2“ there exists a unique holomorphic inverse
branch T‘" of T" defined on B(T”(C"+1),2(§0) and sending T"(C"+’) to 0"“.
Using Koebe distortion properties for the functions T"" and (3.1.12), one can

show that for points y, y’ lying in the same n-cylinder of R2, the following holds:

const
d(y. y') <

”M T” T"’ 1.14
-diamR,,e ( y. y) (3 )

 

Let C,- denote the family of R, such that T(C) = f(R,flX0), i = 1, . . . , 3 (see
equation 3.1.13). We will refer to these cylinders as standard cylinders.

If we select the initial Markov partition to consist of R,- = J ()1,- where each I,-
is an arc of the unit circle that is mapped one-to-one and onto the full circle and the
first of them, I 1, is the arc containing the parabolic point, then the standard cylinders
turn out to be exactly the sets ffnlj, j = 2,. . .,s, where ff’ denotes the inverse
branch that sends the unit circle onto 11.

The set X0 that we get for this particular choice of cylinders is J \ U,, f ‘"(p).
We will denote it by J,.

A similar convention will be made if we work with J f on the real line instead of

the circle.

3.2 Construction of Invariant Measures.

Let 6 = Hdim(J). We will first construct an invariant measure whose Hausdorfl
dimension is equal to (5. To do that we need to introduce conformal measures. Recall
that a probability m is said to be 6-conformal with respect to some transformation
F if

m(F(A)) = [A |F’|"dm for all sets A such that FIA is one-to-one.

44

It was proved in [DU2] that for parabolic systems there exists exactly one 6-
conformal measure m with respect to the jump transformation. Moreover m is non-

atomic.

Lemma 3.12 There exists some K < 00 such that

1
E —— SKforalleX.
IT’(y)|‘s 0

yET‘lx

Proof: Let us denote L(x) = Ey€T_I$|—T,(’W;. Lemma 3.8 implies that if

y, y’ E 0,, for some standard cylinder 0,, then

—1 [T’Wli
B S IT’(y’)|

 

SB

So if x,x' E f(R,flJ,), then fig); S B. Use the fact IT’I" is the Jacobian of the

d-conformal measure with respect to T. So

— I —6 m = m =
ff(R,flJ.)L($)dm($)“Cg/CITQDI d (Ty) 026; (C) 1

But the integral majorizes the quantity Bsup,,Ef(Ri UL) L(x)m(f(R, H J,,). Thus

1
LCD) S Bminlgigs m(f(Ri I] Ll) : K < 00

 

Note that we can actually show that the convergence in the series ZyeT_1, W is

uniform in x E J,, in the sense that

1 to
sup 2 C. W = 6 —+0 ask—>00 (3.2.1)

xEJ. yET—lx m(UZkH ')

Recall that if two points y,y’ are contained in the same cylinder then

45

c’1 S [$71533 S c. That means that as x varies in J,, the sums in (3.2.1) differ
at most by a factor of c, therefore the convergence is uniform.
Let d(x) 2 —6 log |T’|(x), for x E J,,. The measure m, the transformation T, and

the potential (1) satisfy all Walters’ conditions (3.1.1)-(3.1.6). We conclude that

Theorem 3.13 There exists a unique T—invariant probability measure it absolutely

continuous with respect to m. Moreover ,u is ergodic.

For the harmonic measure, fix a reference point P0 on the real line outside the

Julia set and let w = w(-, P0), G = G(-, P0). We introduce J(z) = %5)D, z E C. By

Harnack inequality, for z in a certain neighborhood U of J,
0<cSJ(z)SC<oo

It will be more convenient for us to change coordinates and consider J f a subset
of the real line rather than of the circle, and the parabolic point to be p = 0, rather

than p =1.

Lemma 3.14 1. For all x E J; there exists the limit

lim J(z) = J(x)

z-)x,zEU\Jf

2. For to — a.e. x E J, ,J(x) = Jac[,(x).

Proof: The first part of this lemma follows from Theorem 2.5. For the second
part apply Grishin’s Lemma (2.7) for the functions u = G o f, v = G whose Riesz
measures are 11,, = A(G o f) and u, 2 AG 2 w. We get that for sets E C Jf on

which f : E —> f (E) is injective we have

w(f(E)) = ME) = [E some) = [E J(x)dw(:v)

46

therefore J is the Jacobian of the harmonic measure.

In what follows we mean by Jac,{,(x) this continuous representative J(x). Let

JacuT," be the Jacobian of the harmonic measure with respect to T".

Lemma 3.15 Let C(k1,. ..k,,) be a n-cylinder of R,. The map
T" : C(k1,. . .k,,) —> X0 is injective and ify E T_"x, y' E T‘"x’ are two points of

the same n-cylinder, then

Jacwn (x)

_— — 1
JacZ"(x’)

S c dist(x, x')B

 

 

Proof: Let g be a holomorphic inverse branch of T" defined on B( f R,, 260) =
B(Gkn,260) such that g(T(Ckn)) = C(k1,. ..k,,). Thus g(x) = y, g(x’) = y’. Such a

branch exists by Lemma 3.11. We have to estimate the following ratio:

J(y).7(Ty) . - . J(T""y)
J(v’)J(Ty’) - - -J(T"‘1y’)

 

(3.2.2)

Let us choose points 5, 5’ in U \ J extremely close to y, y’, and let us estimate

G(T”C)Cv’(C’)
G(C”"C’)G(C)

 

(3.2.3)

If we can estimate the previous ratio uniformly in 5 —+ y, 5’ —> y’, we get the estimate
for the ratio in (3.2.2).
Now let 2 = TI, 2' = Tn5’. Both points can be assumed to be in B(Ckn, 260)

as 5 ~ y, 5’ N y’. Rewrite the double ratio in (3.2.3) as

G(92') , G(92v)
G(z’) ’ G(z)

 

(3.2.4)

47

The functions 11 = G o 9, v = G satisfy the conditions of Theorem 2.7, so the latter

double ratio has the following estimate with constants independent of u, v, and z, z’ E

ELI-(Rt)? 60):

 

 

u(zl) , u(z) _ C Z Z, 5
I’U(Z’) ' v(z) 1] S [d( 7 )1

Just let 5 -—> z, 5’ —+ z’, z —> x, z’ —> x’ and we get the desired estimate.

Now to imitate the results of Section 3 of [DU3] we need the following notations:

Ho 2 {g; g : X0 —> R bounded} and ”Ho, 2 {g 6 H0; v,,(g) < 00},

x — x’
p(W’)“

 

 

where va(g) = sup{ , x,x’ E J,,p(x,x’) S e‘1 }.
These are Banach spaces, Hg with the supremum norm and Ho, with the norm

|| 9 lla=ll 9 Ho +1230?)-

Let Ip : J, —> R be a function bounded from above and satisfying v0,(<p) < 00.

For y E J, let

Sn (:1) = v(y) + «p(Ty) + - - . + v(THy)
Let 4.00100)? 2.3-1.. em) My). Then LiIIIIXas) = 2.3—n. eSHWIIIIy)

Lemma 3.16 Iffor some function go as above the operator [1,, satisfies:
H £3 HOS M Vn 21

Then:

|| £39 HAS Mew" || 9 Ha +0 II 9 Ho, V9 6 Ha

Proof: Repeat word by word the proof from [DU3]. Let x, x’ E J, such that
p(x, x’) = e‘k, for some k 2 1. By the definition of the metric, they are in the same

k-cylinder. Choose y E T’"x and let y’ E T‘"x’ be in the same (n+k)-cylinder with

48

 

y. Then
|9(9)- 9y’()l<l|9||e “(W (3.2.5)
657.9(9)
S" (y)_ Sn ) _ Sn ( )
'6 ’p 6 MI €5.1pr 1'6 M <
S C II <9 Ila 6”“kes"“’(”" (3-2-6)
Thus
532“” -£th9'> = Z [63"“0‘y’9(9) -eS""”"/)9(9')l =
yeT—nx
Z esnso(y)[g(y)_ my, 2 g(y y’)[esn‘P(y)_eSncp(y)]_
yET‘M: yeT—nx

The second sum is estimated by (3.2.6) as follows :

C || 99 Ha ll 5" 9 Ho 9(9 33')“ S 6M || <9 Hall 9 Ho p(aw')“

The first sum can be majorated by
6”” ll 9 Ha p(fmr')“ ll £31 lloS Mew" ll 9 Ha p(xfl')“
Lemma 3.17 If the function w denotes — log JacT ,then the operator A, satisfies:

ll 5" 9||oSM “9 Ho

Proof: The operator A, being positive, it is enough to prove that I] £31 HOS M.

Using the definition of £¢, one gets:

531(2) = 2 m (3.2.7)

yET-"x

As in the proof of Lemma 3.12 one can majorize the right side of (3.2.7) by

49

=M<oo.

 

B
minlSiSs W(IUL‘ n J«))

Lemma 3.18 The image under [1,, of any bounded subset of ”HO is relatively com-
pact in ’Ho.

Proof: Before actually proving this lemma we would like to point out that
bounded sequences in ’Ha may not have any convergent subsequence in ’Ho (Arzela-
Ascoli’s theorem does not apply because the space (J,, p) is not compact). Consider
for example the sequence X0"; it is bounded in H0 and has no convergent subse-
quence in the supremum norm. However, after applying £¢ the values of an are
”redistributed” over J, and multiplied with weights 6’” that decrease to zero as the

cylinders 0,, get closer to the parabolic point.

Let (gn),,20 be a bounded sequence in Ha, [I g,, HOS M. We want to show that
one can find a function g 6 ’Ho and a subsequence 9", such that [I Eygn, — g ”0—) 0.

By Lemma 3.17, the functions ['an are uniformly bounded in ’Ho by some con-
stant M1.

For x,x’ E J, we have:

|£w9n($) — £w9n($’)i : i Z lemmgnU/l — e¢(y')gn(y’)] i S
yET‘lx

S X |€"‘y’-e"‘”"||9n(9)| + Z €”“’"|9n(y)-9n(y')|S
yET—lx yET'lx

SH 529'. Ho p(x, 1")“va(v) + 9($,x')"va(9n) M £21 lloS 09(x,$')°

Therefore the family 539,, are equicontinuous on J,,. By the Arzela- Ascoli theorem
for the sets UL, C,, we can extract subsequences (93),,” such that for each k and

for some continuous function g

sup Ifiygfi—glaO asn—>oo
IEUf=1Ci

50

Consider the diagonal subsequence (g3),, and relabel it g,,. We get that for each k

Q.

8:l sup [[Iwgn — g| -—) 0 as n —> 00 (3.2.8)

IEUle Ci

k
611

Finally, to show that the subsequence Egg" is uniformly Cauchy, let 6 > 0 be fixed.

I£ign(x) -£i9n+p(x)| S X e“"y)l9..(v) -9n+p(y)| S

yET‘lx
Z 6"(y’l9n(9) - 9n+p(9)| + Z ew‘”’l9n(9) - 9n+p(9)l S
yET'll‘rMUle Ci) yET“IU(U:k+1 C‘)
S 6i: ii £411 “0 +(ii 9n ”0 + I] gn+p ”0) sup 2 812(9)

xEJ. yeT—lxn(U:k+l Ci)
If we denote by 6k the supremum in the previous inequality, by an argument

similar to the one in (3.2.1), we have that 6k

—> 0 as k —+ 00, so for some large
k = k(e), 6k S 6/2 . For this fixed k there exists some index no such that

6:: S 6/2 Vn 2 no. Therefore [I £,,,g,, — £,,,g,,+,, HOS e Vn 2 n0 2 n0(e)

The last three Lemmas allow us to use a result of Ionescu Tulcea and Marinescu
[ITM] which states that 5,, on the complexification of H0 can be decomposed
into the sum .6, = P + S of a projection P onto the finite dimensional space of
eigenvectors with eigenvalues on the unit circle, and a contraction S of 710.

On the other hand, Schweiger formalism applies for the system (T, J,,w,R,).
In particular there exists a unique T -invariant probability equivalent to w, call it

du = pdw . This measure is ergodic, logp E L°°(w), and the following holds:

/ gpdw=/ goT"pdw=/ g£3pdw
J. J. J.

This computation shows that p is an eigenvector of .C with eigenvalue 1 and that

there are no other eigenvalues of the form A = ehi except for A = 1. ( Because if p),

51

were an eigenvector for such a A, then p,\dw would be a T ”-invariant measure, and
by uniqness in Schweiger formalism this means that p), = p.)

The number of eigenvalues in the theorem of Ionescu Tulcea and Marinescu being
finite, we get that it is impossible to have eigenvalues of modulus 1 whose arguments

are not rational multiples of 27r.

Theorem 3.19 Let I/) = — log JacT for the harmonic measure on parabolic Julia

LI.)

set. Then

1. The operator [5,, : L°°(w) —> L°°(w) has only one eigenvalue of absolute value 1,
this eigenvalue is 1, and the eigenvector p is continuous. The eigenvector p is

normalized such that f]. pdw = 1.
2. [3,], = P + S where P is the projection F(g) = (fJ' gdw)p
3. The operator S acts on Ho, and I] S" ||,,S c17[1 for some 71 E (0, 1).

In the sections that follow we will denote by V the T—invariant measure equivalent
to w and by p its density with respect to w. We will denote by p the T-invariant
measure equivalent to the 6-conformal measure m and r will be the density 34%.

Both invariant measures being ergodic by Schweiger’s formalism, there are only
two possible situations: either V = u or V _L ,u.

If V = u then their Jacobians ( with respect to T ) are equal so
logJ$+p—p0T=logJ,:+r—roT

Thus
log J3; — 510g |T’| = 7 o T — 7 (3.2.9)

where 7 is a bounded function. We use an observation from [DU3] that the homology

equation (3.2.9) for T implies a similar homology equation for f.

52

Recall that T(x) = fN(I)(x) for some N(x) Z 2 then N(x) is constant on

cylinders and

N(x)— 1
log .13," = long(f 1))
i=0
N(x)—1 .
élong'($)|=5 Z |f'(f’$)|
i=0

Therefore if we denote by O(x) = log g(x) — 610g |T’(x)| we get that

8(x) — O(fx) = log Jflx) — 610g |f’(x)| (3.2.10)

On the other hand (3.2.9) for T implies that

9(117) = 7(fN(x’($)) - 7(2), GUS?) = 7(fmx’ (1%)) 7(fx) (3.2-11)
Combining (3.2.10) and (3.2.11) we get
log Juf,(x) — 610g |f'(x)| = 7(fx) — 7(x) a.e. x E J (3.2.12)

Since the left side of the equation (3.3) is continuous and the tree of preimages
UnZO f ’"(x) of every point is dense in J , one can follow the proof in [PUZ] to show
that the function 7 has a continuous representative and that the equality (3.3) holds
everywhere in J.

The next section contains a brief explanation on why (3.3) leads to contradiction

(therefore only 02 J. m can happen).

3.3 Solving the Homology Equation.

This section is a compilation of results from [PV] where it was shown that the homol-

ogy equation leads to a contradiction . Most of them are intermediate steps toward the

53

construction of new coordinates (in a neighborhood of some reppeling periodic point)

that simultaneously conjugate the dynamics to a linear map and to a translation.

v(2) ’

Throughout this section we use the notation 3%] = lim,_,,, x E J, for positive
harmonic functions u, v in N \ J, vanishing on J (here N is a neighborhood of J).

First we are going to introduce some notations: let I‘ be a cone with the vertex at
the parabolic point p, containing J, such that the forward orbit of the critical points
does not intersect I‘. for x E J Let D(x) denote the largest disc centered at x and
contained in the cone I‘ ,and let I’, = c0nv(p, D(x)). Denote by f,— 1 the inverse
branch of f that sends the parabolic point to itself. All the functions ff” are well
defined in F .

For a set E we will denote fan by E_,, .

The homology equation can be written as follows:

log G32) — 610g |(f")’(:v_..)| = 7(x) — was...) (331)

 

Lemma 3.20 If the homology equation (3.3.1) holds for some continuous 7, then
V2 6 F \ J

3 the non-zero and finite limit "1320 |(f")'(z_,,)|" G(z_,,).

Proof: Let us estimate bag—23%) where fin déf |(f")’(z_n)|" G(z_,,).

Let x be a point of J closest to z that contains 2 in D(x). Then:

 

 

 

fln+k(2) _ O flunk?) , 931(4) 0 5n+k($) 2
’3 no) ‘1 i932) ' (W) i ,1, am

G($—n—k)
G(.’L‘-k)

 

: 10g [571042) , fin+k($)

ac) ' 4.0)] +10,

i(fn),(x—n—k) '6 =

54

 

_ log 16n+k(z) , fin+k($)

_ 33(2) ’ 51:01?) + 7(’E_n—k)—7(Lk)

The last term tends to zero (uniformly in n and z) when k -—> 00. The first term can

be written as:

G(z—n—k) _ G(x—n—k)

(fk)’(z—I-) , (f’°)’(x—1)
10g[G(Z—k) ’ G(Lk)

(f n+k),(Z—n—k) ' (f"+")’($—n—k)

 

]— 610g|

 

Let us consider the first term of the previous sum; it can be written as :

 

oras

 

. dISt(z_ ,x- ) v 1
Use W A E ——> 0 ask —) 00 to apply Theorem 2.5 for

u = G o ff", v = G in the domain D(x_,,). We get that

 

(G 0 ff")(Z-Ic) G(z_,,) 1
lo : < c— —) 0
g’(G o from» Gas-1.)] - A"
uniformly in n and z.
The second term can be written as
610g] (flu) (IE—H S c1 —) 0

(ff )'(Z—Ic) k

uniformly in z and n as ff" are univalent functions and W x %.

This is enough to conclude that the limit exists.

Let us denote v(z) = lim,Hoe |(f")’(z_,,)|6 G(z_,,) for x E I‘. As the limit in

55

Lemma 6.1 exists, we conclude that

v(fz) = v(z)]f’(z)|’ if z,f(z) E F. (3.3.2)

The function v is not harmonic.

Let pF denote a smaller cone with vertex at p.

Lemma 3.21 In the cone pl‘ the following holds:

 

 

 

—1 v(z)
< <
C” — G(z) “ C”
Moreover, for x 6 J01" ,
v(fiv) . v(Z) -
G(x) :2; 0(2)

Proof: It is in the spirit of the previous proof. The interested reader can find it in

[PV].

Notation We are going to denote vp {if v and we will call it ”parabolic point
automorphic function”. We will introduce another function of this type and show
that they are proportional.

Choose x0 to be a repelling periodic point of f and let it be close to p such that

orb(Crit(f)) flD(x0, Ixo — pl) = 0.

Let I be the period of x0 and let (f’)’(x0) = A > 1.
Let F = f’ and F ‘n be the holomorphic inverse branches that send x0

to x0. These functions are single-valued holomorphic functions in the whole disc

D($01|$0 _ pi)

56

Lemma 3.22 For all z E D(xo, |x0 — pl) \ J,
3 the non zero and finite limit ”1320 |(F")'(F'"z)|6G(F—"z).

Proof: Let us estimate log[bn+k(z) : bk(z)] where b,, “if (F")’(F‘"z)|"G(F‘"z).

Let x E J be the point closest to z. Then z E D(xo, |x0 — p]) and

log b,,+k(z) = log[

bn+k(z) _ bn+k($)] 1 bn+k($) :
bk(Z)

bk(Z) ’ bk(x) 0g 0],;(1‘)

 

 

 

bn+k(Z) . bn+k(x)
bk(Z) ’ bk($)

__ bn+k(z) . bn+ ((13)
—log[ bk(Z) ’ bk???)

G (F ‘n’kx)
G(F—kx)

 

= log[ ] +103 l(F")’(F"’"°$)|" =

 

l + (NF—"4%) - Alf—"13D-

The last term tends to zero when k —> oo uniformly in n and in
z 6 pD(xo,|x0 —p|) for every p< 1.

The first term can be written as

g(F—n—kz) . G(F-"—k.’r) (Fn+k)l(F—n—kz) . (Fn+k)l(F—n—lcz)

 

 

 

 

 

log[ G(F'kz) . G(F‘kx) ]+ 610g] (Fk)’(F"‘z) . (Fk)’(F""x) | (3.3.3)
Let us rewrite it as:
log[(G o Arr-2(2) , (G o F'""‘)(:v)] z
(G 0 F499) ’ (G 0 F"‘)($)
10 [(G’ o F‘")(F""z) . (G o F‘")(F“"x)]
g G(F—kz) ' G(F—kx) '
But if z E pD(x0, |x0 — p]) then
dist(F"°z, F“"x) S CPA—k (3.3.4)

 

[$0 — Pl

57

Considering u = G o F“", v = G in D(xo, |xo — pl), from the Holder continuity of
log% we conclude that

u(F—kz) . u(F‘k )

x
l ' . < , A"’"’ 0 k
I 09, v(F‘kz) v(F‘kx)’ — C100) _I as —+ 00

 

uniformly in u, v, z when 2 E pD(x0, lxo — pl) . That means that the first term of
(3.3.3) tends to zero when k —+ 00 uniformly in n and z E pD(x0, lxo — pl).
The second term of (3.3.3) can be written as

(mm-Hz)
(F")’(F‘"”‘I)

(F ” ’(F‘kx)

— )
6108 l (F—n)I(F—kz)|

 

|=6log|

 

Now (3.3.4) and Koebe’s principle applied to the univalent function F 'n in

D(xo, lxo — pl) Show that

 

610g|( ‘")’(F"‘ >

(;_n),(F_k:)| S c2(p)A_’c —> O as k —> 00 (3.3.5)

where the convergence is uniform in n and z E pD(x0, lxo — pl).

Remark: Let us denote v,0(z) = lim,,_,oo|(F")’(F‘"z)|5G(F’"z), for

Z 6 D(SFO, l$0 " Pl) Then
v,0(Fz) = v,0(z)lF’(z)|" for F2 6 D(xo, lxo — pl). (3.3.6)

The function 2),,0 is NOT harmonic.

Lemma 3.23 In the disc pD(x0, |x0 — pl),

 

58

Moreover, for x 6 JD D(IEO, [1130 — Pl)

:: lim vx0(z) : e—7(x)e7(xo)

G(IE) z—rx, zED G(Z)

 

 

 

Proof
920(2) _ ,m G(F‘"2) G( F130) G(F "AOIIF .I___F "_) IzzoI.

The middle factor equals 1 by homology equation. The last factor is bounded in-
dependently of n and z E pD(x0, lxo -— pl), by Koebe’s principle. The first fac-
tor is bounded independedly of n and z E pD(x0, lxo — pl) by Theorem 2.5 for
u=GOF"”, v=G in D(x0,|x0—p|).

Let 2 approach x E pD(x0, lxo — pl), for some p < 1.

 

 

GUS—"2) G(F""III) G(F""$) a (F ___"__)’($ ) a
=lim : F"’
The middle factor equals (“Helm—"3) and thus tends to e‘7($)e'7(‘”°) uniformly in
z E pD(x0, |x0 — pl) because x_,, E pD(x0, lxo — pl) when n —> 00.
The logarithms of the first and the last factor can be estimated uniformly in n by

c(p) |x — zl" and by c(p)|x — zl respectively, using Theorem 2.5 and Koebe’s distortion.

Cl

Let us consider the following functions:

2 2 [log I f’(z_,,)l — log lf’(x_n)ll

n21

where z_,, = ffnz and x_,, = fl—"xo for z E I‘, and

3200? =2 [lole'(F "Z)| -108|F'($o)|]

n>l

59

for z 6 D(xo, |x0 — pl). These are symmetric harmonic functions in the domains
I‘, D(xo, lxo — pl), respectively.
Consider

h,,(z) “’2 / e-WZWWII dz (3.3.7)
550

and

h,0(z) déf/ e_([’10(z)+’fl}’°(2)) dz. (3.3.8)
$0

Due to the symmetry of 3,,(2) and fi,0(z) one can choose their complex conjugates
such that hp and h,0 are real on the real axis.

Because

flp(fz) - 5143) =108|f'(Z)| for z,f(Z) E F
5,0(Fz) — 510(2) 2 log |F’(z)l — logA for z,F(z) E D(xo, lxo —p|)

we have that:

 

 

h’ (fz) e‘BPW) 1
hp(z) 6 W2) If (le
h’0(Fz) A
' hioIzI ' IF’(z)| Or Z’Fz e (”3” "B" p” (3 3 0)

Let us consider Koenig’s function (or F atou coordinates) (13,0:

 

It is well defined in D(xo, lxo — pl). See [CG] for details.
We can see that <I>;0(z) = h;0(z) and that both (1)30 and h,0 vanish at x0.

Therefore

60

Similarly, if
_ - Ill—”(2) — fl—n(370)
‘1’”(3’ ‘ 332° (ff")’(:z:o)

then (1),,(x0) : 0, and 4);,(2) = h;,(z). Therefore (1),, E hp. In particular

 

h$0 o F(z) = A hx0(z) for z,Fz E D(xo, lxo — pl), A > 1 (3.3.11)
hp 0 f(z) = hp(z) + a for z,fz E F, a > 0 (3.3.12)
hp(330) = (1200130) = 0: hp(P) = 00 (3-3-13)

Let us notice also that

0,,(2) = e554” lim G(z_,,)l(f")’(x_,,)l6 for z e r

11—900

v$0(z) = 6663,,(2) lim G(F‘"z)l(F”)'(x0)l6 for z E D(xo, lxo — pl)

71—)00

In particular the limits in the right part exist. Let us denote them by fp(z) for
z E I‘, 7,0(z) for z E D(xo, lxo -— pl). These are positive harmonic functions defined
in I‘ \ J and D(xo, lxo — pl) \ J , respectively, and subharmonic in F and
D(xo, lxo —— pl) respectively, and vanishing on J.

By the first inequalities of Lemma 3.21 and Lemma 3.23, if p < 1, then:

 

2:]: x 6‘55P(z)cp for z E pF
:35?) x e—6Bx0(2)6p for z E pD(£E0, [$0 — Pl)

Use the equalities in Lemma 3.21 and Lemma 3.23 to derive:

 

T1417) : e—éfip(x)e-7($)67(P) for x E pI‘fl J

£2
a

61

”7,0 (1‘)

~55: (x) —7(x) 7(xo)
=6 0 6 6 fOI‘TEpDIB,IE— IIJ

Denoting 7,, = 7,,(z)e‘7(”), 7,0 = 7,0(z)e‘7(‘”°) we conclude that

, 6
u(x)

TP(:L’)
7.150(3)

 

 

 

If we introduce

we see that A is a well defined holomorphic function in F f) D(xo, lxo — pl) and

”(3’) = |A(x)l2 (3.3.14)

711:0 (SC)

 

Applying Lemma 2.8 on harmonic rigidity (according to [A1] or [DU3] the Julia
set has linear measure zero ) and its corollary for the functions 71,, 7,0 that satisfy
(3.3.14) in I‘flD(x0, lxo — pl), we get that the two functions are pr0portional. On

the real axis

hf)($0) l _ e‘flpml _ 1
hfro($0) — 64M“) —

 

 

 

2] = const =|

so actually 7,,(z) = 7,0(z) d-if T(Z) for z E I’flD(x0, lxo — pl).

So h;,(z) = h;0(z) and hp(z) E h,0(z) in I‘flD(x0, lxo — pl) because h,,(xo) =
h,0(x0) = 0.

Let h(z) déf hp(z) = hxo(z) for z E I‘flD(x0, lxo — pl) . The functions
hp and h,no being defined as Fatou coordinates are univalent and the following con-

jugations (linearizations) hold:

(1(2): AMP-52) 2. F'1z E D(xo. l$0 — pl)

62
h(z) = h(fl’lz) + a a > 0,z,f1"z E I‘
h(xo) = 0, h(p) = 00 (3.3.15)

Consider the domain 0 = h(I‘flD(x0, lxo — p|)) and let R(w) = T(h‘lw), w E O.
The map R is defined on O and

R(Az) = /\"R(z) for z, A2 6 o

R(z-I-a) =R(z) for z,z+aE(’)

Extend the map R to the whole plane using the first equation and denote the
extension by R1. Extend the map R using the second equation and denote this
extension by R2. Being holomorphic extensions to C of the same function, R1, R2
are equal.

In particular the zero set of R is invariant under the group of transformations
generated by g1(z) = A2, g2(z) = z + a , which contains elements g,, = z :t en, for
some 6,, ——> 0. So as R(w) = 0, then lemn = 0.

But this is impossible. Just take a point y to be one of the endpoints of a
complementary interval of h(J). Then R(y — en) is positive starting with certain

n. We come to a contradiction.

CHAPTER 4

Thermodynamical Formalism for

Countable State Systems

4. 1 Entr0py

Let ,u be a T invariant probability measure on J,. Let Q = {(21, Q2, . . . ,Qk . . .} be a
finite or countable partition of J,. Denote by Q = Q,l . . . , Q," the set

62,-, flT‘lQ,2 fl . . . (WT—"+162," and by W,,(Q) or Q V T"1Q V . . .T‘"+’Q the set of
all such words.

Notation: Let H,,(Q) “’2 2,740,.) log 745—0-

Lemma 4.1 For any partitions Q and Q,
HAQ v Q) s HIIQ) + HtIQ).

Proof: See Lemma 1.17 in [B0], page 28.

Lemma 4.2 If Q is a finite partition of J, then there exists the limit
1 e
3320 5H,,(Q v T-lov . . . v T‘"+‘Q) d=f H,(T, o).

63

64
Proof: See Lemma 1.18 in [B0], page 28.

Lemma 4.3 If Q is a finite partition, then for any k,
H,,(T, Q v T‘IQ v . . . VT‘k+1Q) = H,(T, Q).

Proof: See Lemma 2.2 in [B0], page 46.

Definition The entropy of the measure u is defined by
def
MT) = SUP HAT, Q).

where the supremum is taken over all finite partitions Q of J,.

Note that the entropy may be equal to zero or infinity.

Lemma 4.4 Let B 2 {B1 . . . , B,,,} be afinite partition of J, and let 1),, be a sequence
of partitions with diam(D,,) —+ 0 as n ——) 00.

Then there exists a sequence of partitions 8,, of cardinal m, 8,, = {Elfi . . . ,E,’,’,}
such that each E? is the union of some elements of ’D,, and
lim,,_,oo u(El’ABJ = 0 for each i = W.

Proof: As in Lemma 1.23 from [Bo], page 33, except that the partitions ’D,, may

be countable.

Lemma 4.5 Let c > 0 and let 8 be a finite partition of J,.
Then there exists 60 > 0 such that for all partitions D with diam(D) < 60 the

following holds :
HII(BV D) — H#(D) S 6

Proof: As in Lemma 2.3 from [B0], page 47.

65

Lemma 4.6 Suppose 1),, is a sequence of finite partitions with diam(’D,,) ——> 0. Then

It,,(:r) = lim H,,(T, 7),)

‘11—’00

Proof: See Proposition 2.4 in [B0], page 48.

Theorem 4.7 Let ,u be a T-invariant probability measure on J, such that

 

1
;#(CI) 108 M01) < 00,

where C, are the standard cylinders from the end of section 3.1. Then the following

limit exists

lim -1—H,,(C v T‘lc v . . . v T‘"+IC)

11—)00 TI.

and it is equal to h,,(T).

Proof: By the T-invariance of p , the sums E,p(T"‘G,) log p—(TT’FCT) are finite
for each k, and the sequence %H,, (C V T‘IC V . . . V T‘"+IC) has a finite limit which
we will denote by L.

To show that h,,(T) S L , let D9, = {01,02, . . .,C,,,U,,Z,,+1C'k}, and
1),, = D9, v . . . v T-"+1D3.

Then diamDn -—> 0 and

lim h,,(T, 139,) = "1321071,,(Tpn) = h,,(T). (4.1.1)

n—->oo

Together with H,,(T,D?, V V T‘Hl’Dg) S H,,(T,C V . . . V T‘kHC) , it implies
that h,,(T,Df’,) S L and therefore that hu(T) S L.
For the reversed inequality we will show that for any 6 > 0, h,,(T) Z L — 6.

Claim that if two partitions of J,, B = {B1, . . . Bm, . . .} and

66

.7: = {B1,. ..Bm,J*\UZ‘=1Bk} satisfy
HH(B) S Half) + 6 (4.1.2)
then for an arbitrary partition A of J,., the following holds:

Hp(BvA) g H#(J=VA) +e

To prove the claim let B = J,.. \ UZ‘ZI Bk. By (4.1.2) we get

 

 

B,- 10 —- — B 10 — S 6
film ) g “(31.) u( ) g u(B)
and
°° u(B)
Bi 10 S 6.
(i221 #( ) g ”(31)
#(BnAa)
A ) i— 1.2:] z(12(1-aflgll(BiflAa)
°° ”(Bin/4a) u(BnAa)
: Bi Ba —-——-10 _—
i=rEn:+1 M ) H(Bi) g#(BiflAa)
By the concavity of the function :1: H loga: we get :
H#(BV.A) — “()f’V/l S2 ,u(B logZflBnA
i: m+1
°° 11(3)
2 Bi 10 S 6.
£331,“ ) g ”(Bi)

Therefore the claim is true.
Fix some 6 > 0. Then for all large enough n, the partitions D2 are fine enough

to ensure that

Hu(C) S H,‘(’D2) + e for all n 2 n6.

67

Therefore Hp(T‘kC) S HAT—hug) + 6. Applying the claim k times we get :
H#(C VT’IC v . . . v T“"C) g HAD}; VT"‘C v . . . vT‘kC) + e g

H#(ngT-1ng...vT-k6) + 26 g
g H#(D2VT_1D2V...VT"°’D2) + k6
=> %HM(CVT‘ICV...VT"‘C) g %H#(D2VT‘1D2V...VT"°D2) + 6
=> L g h,‘(D2) + e = Mon) +6

=>LSh#(T)+6. C]

4.2 Pressure

We will define the pressure for potentials (b in the space of functions

71’ add {(51 J —+ R ;¢ is bounded from above, and va(</>) < 00 }

For such functions (15 define :

Snob

I:'U

1:161:¢(CF )for a n — word A 6 mln (C) (4.2.1)

and

Zn(¢>) = 2 6X1) Sn¢(_A_) (41?)

AEan)
Remark For functions (15 E ”H; the sequence filog Zn(gb) is decreasing . Define

the pressure to be

Pure) dé’ Po) "’“” lim —logz (a5)

n—mo n

68

It can be a real number or it can equal —00.

Theorem 4.8 Let (b E ’H; and let u be a T-inuariant probability measure on J,.
satisfying Hp(C) < 00. Then

hu(T)+ Acid/4 3 13(4) . (423)

Proof: If f (bdu 2 —00 there is nothing to prove. Assume that f gbdu is finite. We

will show that

AEWn(C)
— 1(2 u(A)log-1—- + / S444) —
” 46mm) — “(A) J" n
= %H,.(cvr-16v...v:r-"+IC)+ [J ¢d,u spas) (4.2.4)

Let 24 E A satisfy : f4 anbdu) S u(A)Sngb(zA). Then

1 1
5 Z (u(amgfl; + Law) 3

1 exp Sn¢(zA)
< — A l <
__ n gm u(_) 08 #(A) _

1
S —log( 2 exp Sn¢(ZA) ) S
" newnm
1 1
S —log( 2 supexp an’)(z)) 2: —Zn(gb) (4.2.5)
" aewnw) A "

=> h..(r)+ / <15 du 3 Po)

Let 5 = Hdim(J).

Theorem 4.9 The function (b = —610g |T’| has P(q’>) = 0.

69

Proof: Let A E Wm(C). The map Tm : A —> J,, being one—to-one and
onto, there is a natural identification between Wm(C) and the set T‘mx , given by
A 6 Wm (C) <—> 224 = AnT'mL for arbitrary points x in J,,.

Fix such an x and let $4 = AflT’mJ*.

m—l
Smcbw.) = SUD Z </>(Tiy)
316A i=0

m—l m- 1
g 2 ¢(T :cA) + 2 811p |¢(T':c )- ¢(T‘y)l S
i=0 1_() x ,yEA
<mz¢( MA) + ||¢|| 26"" ”"<
i:0 i— 0
<:1¢( )+ const (42-6)

i=0

Let \11 be the fixed point of the Perron Frobenius operator, £4911 = \II defined in
Section 3.2. According to [DU3], it is bounded away from zero and infinity. By the

inequality (4.2.6) we get :

Smgb()A )S 2% TxA )+const+log\II(:rA) =>
:0

8Xp3m¢(A) S “DMZ ¢(Ti$.4))‘1’($4) =>

Zm(¢)= Z exp3m¢(A)S

AEWMC)

S e 2 exp qub(a:A)\Il(:cA) = c(£m\IJ)(:1:) = c\II(:c).

$AET"’"1:
The function \11 being bounded (see Theorem 3.19), we get : Zm(¢) S M2

and P(¢) = limm_,oo #Zmfib) S 0. The opposite inequality follows identically,

70

using
m—l m—l
811p Z ¢(T'y) Z Z ¢(T‘Iv4)-
yEA i=0 i=0

Theorem 4.10 (Variational Principle) Let (b = —6 log |T'|. The probability mea-
sure u equivalent to the 6-conformal measure is the unique T-inuariant measure that

satisfies:

HA(T)+ [J odA = 0 and HA(C) < 00. (4.2.7)

Proof: Let us prove uniqness first. Let A be a T—invariant measure that satisfies
(4.2.7) First let us assume that u i A. Then there exists a Borel subset of J,., call it
B, satisfying

u(B) = 0, A(B) = 1, and T—IB = B (4.2.8)

By Lemma 4.4 for the partitions B = {B,J* \ B} and
Dm = D9,, V T‘ng, V V T""‘+1D?n where D3,, = {Cl,...Cm,U,‘:°=m+le}, we

can find sets Fm that are union of standard cylinders and that satisfy
(u + A)(BAFm) —> 0 as m —> 00 (4.2.9)
By Theorems 4.8 and 4.9 we get:
0 = P(q5) = inf %H,\(C V . . . V T‘"+1C) + [B (15d /\ S (4.2.10)

3 l(HA(CV...VT‘"+IC) + [Snod A) =>
n B

as Z (A(AMog 1(2) + [403ande

Aewn (C)

 

By an argument similar to the one in Theorem 4.8, if a: E J,. is fixed and $4 =

71

AH T‘":r , then

[3 5.44m 23 4(4)[SA¢(rA)+ II¢IIA1 s

AEWMC)

S 2 A(A)Sn<b(:c4) +c1 :

AEWMC)

 

 

1
—c1+1S A(A 10 ex Sngba: + AA) 10
4;. )[ g((A) p (4)] An§:0( [g (A)

 

2:4an exp 571M354) Zgnnpo exPSn¢($_/1) _
S “Fm" A(F.) + "(J*\F")‘°g 4mm.) ‘

SA(Fn)log[ Z 63""’($4)XF.($.4)1+
AEWn(C)

1
+ A(L \ FA) log [ Z esmwfil XJ*\Fn($A)] + 2 supxlog — (4.2.11)
Aewnw) [0,1] ‘17

By (b E ’H; and by the boundedness of log \I' we get that

m(A) X 65.4%) [Ac—SW”) dm(z) = eS”(“’4-)/ |(T”)’|h dm
_ A

Therefore

esfiflxfi) x m(A)

x
,E
'3:

(4.2.12)

and (4.2.11) becomes:

—C2 3 A(Fnflogl Z 6144) XF.($4)l+
AEWMC)

+ A(JA\Fn.)108 l 2 6M4) XJ.\F.. (113.4) l
46mm)

= A(Fn)10g(/J’(Fn)) + /\(J* \ Fri) 10801074 \ Fn)) (4-2-13)

By u(Fn) —> u(B) = 0 and A(Fn) —-> A(B) = 1 as n —> 00 the inequality (4.2.13)

+Sn¢($fl) l S

72

becomes —c2 S -—00.

Therefore A and u can not be orthogonal. In particular this gives that in the
class of T-invariant ergodic probabilities there is at most one that satisfies (4.2.7).
This will be enough for our purposes.

To prove uniqness for the general case, decompose the measure A into
A = A + [i where A J. u and [i < u and follow the proof of Theorem 1.22,
page 31 in [B0].

In order to prove that the equality (4.2.7) actually holds for the measure ,u it is
enough to show that hA(T) + f(b dA 2 0.

Fix n > 0 . We have:

1 —1 —n+1 __
nH,,(CvT Cv...vT C) +/Jfigbd,u—

 

 

1 1
= gégvnlMA) 1024“ A) + [45.4 dul 2
1 .
Z BA§VA[M(A) log ”(4) + #(A);2£Sn¢($)l (4.2.14)

Using one more time u(A) x e5"¢(“A) = |(T")'(:1:A)|"6 and the distortion pr0per-

ties of T it follows that

 

1 n—l .
log 2 bsup( 10 T’ T's: +c= —infSn¢ :1: +c
144) M Z g1 |( )> ()

and

hu(T)+/J ¢dHZ 0

which conclude the proof.

Theorem 4.11 Let A be a T-inuariant measure with positive entropy satisfying

 

 

73
HA(C) < 00. Then
. hA(T)
Hd A < 4.2.15
"7“ ) - L. long’l dA ( )
Proof: Let o > 112)?) (1A. We want to Show that Hdim(A) g o. Let 60 > 0
J4
satisfy
60 < — — 1 h : 4.2.16
04 W ere “0 fJ.10g|T’| dA ( )
Then there exists some 715 such that for all n > n; the following holds:
1
aHA(C v T‘IC v . . . v T‘"+IC) 3 (1+ 60)hA(T). (4.2.17)

Let 6 > 0. Let n > n.E be large such that diamA S e for all A 6 W" (C)

 

Let .7}, be the family of all n—generation cylinders having positive A mass and let

Then A(X) = 1 and if

HOAX) déf inf{Z(diamB)"; the sets B cover X and diam(B) < 6 }

we get :
1
Ha,e(X) S (diamA)°‘ >1 —— :
A24}. A23... |(T"’(a:A)l°‘
= Z e—aSnUong’lxrA) = Z A(A)e_as"(l°gITII)($4)+1°gl/A(A) S
AEJ-‘n A635}.
_<_ exp 2: {—aA(4)S.(log IT’I(2:A) + 4(4) log 1M4) 1
Aef'n

By Lemma 3.8

(T ”)’($4)

S c for all z E A 4.2.18
(Toe) ' 3 ( )

|5n(10ng'|)($4_) - 51.001; lT'|(Z))| = |

74

Therefore

 

HA,.(X)Sexp( Z [—a/ Sn(10ng’l)d/\ + A(AHog 1

Aer-n A (4)] ) 863 S

s c. (explln‘i [X Sn(10ng’|)dA + fig A(A)log—(:4_—)l)" s

S 04(exp[—oz/X log(|T’|) dA + (1+ 6)hA] )"

By (4.2.16)
—a/Xlog(|T’|) dA + (1 +60)h,\ = —a/Xlog(|T’|) dA + (1 +60)ao/Xlog(|T'|) dA

= —[a — 00(1 + 60)]flogIT'| dA

The jump transformation being expanding and the way 60 was selected imply that

—[a — 00(1 + 60)] flog |T’| dA is negative, say equal to —61. We get
HOAX) S c4 ("6‘ for all n _>_ n.. (4.2.19)

Therefore Hdim(A) S a for all a 2 W. This is enough to conclude that
- h

Lemma 4.12 If u is the T-invariant measure equivalent to the

6 conformal measure m, then

1
CA 10 -———<oo
gm ) gum")

Proof: Without loss of generality we can assume that inf J = 0 and that sup J = 1.

75

The measures m and u being boundedly equivalent, it is enough to show that

 

1
m(C n)log < 00
£3. mm.)

Use the fact that |T’I‘5 is the Jacobian of m to get that an |T’I5dm = 1. By

 

T’ ,
Mfl S |T’(:’))| S M1 for x,x’ in the same cylinder (4.2.20)
we get that |T’(:r)|6>< m(—C— ) for all a: E C".

On the other hand if the cylinder C" is Cn = [am bn] D J, then

/: |T’(:1:)|d:z: = /: T’(:r)dx = T(bn) — T(an) = 1 — 0 = 1

so according to (4.2.20) we get |T’(:c)| X '01". for :1: E C”. ( The notation |E| means

 

the diameter of the set E.)
By construction |Cn| = IffK"I| for some interval I that is mapped by f univa-

lently onto [0,1], where Kn is the integer part of Using the estimates in the

_n_
deg(f)-1 '
Appendix we get

, 1 1
|T(:c><)| T—Cnl x—EforzECn

so m(Cn) X "—125. The measure m being finite we get that 1 x 2,00 ”—125, which implies

that 26 > 1. Therefore

 

1 1
Zm(C )log M(0)Sconst+2— 610g —2—n6<oo

n>0 n>0n

for 26 >1.

Lemma 4.13 If 1/ is the T~invariant measure equivalent to the harmonic measure

76

to, then

 

1
Z u(Cn) log u(Cn) < oo

n>0

Proof: It is enough to show that Zn>0w(Cn) log < oo.

__1_
w(Cn)

As in the previous proof we will use the fact that lCnl x # and lxl X i for

:1: 6 Cu to estimate the harmonic measure w(Cn). Let inf CA 2 an, sup C" 2 bn, on =

an+bn _ bn—an
2 i771“ 2

Let us consider the auxiliary domain (2,, contained in C \[0, 00) whose boundary

coincides with

89,. n B(cn, rn) = JflB(c,,, rn)

69.. fl( [0, 00) \ B(Cm rn) ) = [0, 00) \ B(cn, re) (42-21)
Let tog" be the harmonic measure in the domain 9". It is clear that
MC", ) 2 wn.(Cm -) (42-22)
The Julia set J being uniformly perfect gives

an(Jfl B(cn,rn),y) Z c(fl) for Vy E B(cn, gr")

for some constant c(fl) depending only on the constant 6 from the definition of
uniformly perfectness of J. In particular if w denotes the harmonic function in C \
[0, 00) that vanishes on [0, oo)\B(cn, 313m) and which equals 1 on [p, 00) fl B(cn, %r,,)

we conclude that for every y E C we have:

wn.(J fl B(Cm re), 9) 2 6(3) 112(9) (42-23)

77

The inequality (4.2.22) becomes

w(Cn, oo) 2 an(Cn, oo) 2 c(fl) w(oo) (4.2.24)

Using the change of coordinates z ——> Z2 the upper half plane is mapped onto the

domain C \ [0, 00) so one can easily estimate

w(oo) Z crncgl/2 x n‘3/2

Finally this computation together with (4.2.24) gives

 

w(Cn) 2 cn‘3/2 2 C’lxl3/2 v2: 6 0,, (4.2.25)
which implies:
Z w(C,,) log 1 S E w(C,,) inf log _lT/z + c.
n>0 W(Cn) n>0 xEC" lxl
So
2:an )log— )S c"/J loglz: )+c'— — d’uw (0)+ +c’
n>0

where (J,,, is the potential of the harmonic measure. The Julia set being uniformly
perfect, it is regular for the Dirichlet problem; as a consequence the potential Ll“, is

finite everywhere on J. In particular Llw (0) < 00.

Applying Theorems 4.8, 4.10, 4.11 we get that if the invariant harmonic mea-
sure and the invariant conformal measure are orthogonal, then the dimension of the

harmonic measure is less than the dimension of the Julia set.

CHAPTER 5

Appendix

The next proposition is a particular case of Proposition 8.3 of [ADU], where all
necessary references are made to restore its proof. But we decided to include the
proof here to make the thesis as self contained as possible for the convenience of the

reader.

Proposition Let f be a rational function with parabolic point p = 0 whose
.11 ulia set J j is contained in R and whose only Fatou component is the parabolic
basin.

Let f1“ 1 denote the inverse branch of f that sends the parabolic point to itself.

Then for any closed interval I C J f close to 0

1
n

1
|ff"I| x B— and V11: E ffnl Ice] x (5.0.1)

2

Proof: It is easier to carry out the estimates if we change the coordinates such that
the parabolic point becomes 00. If we denote the new function g without loss of
generality we can assume that (0,00) C F9, 0 E J9 and that the expansion of g near
infinity is

A B
g(z)=z+1+—+—2+...
z z

78

79

or simply g(z) = z + 1 + f + 01(2) = z + 1 + f + 2% + 01(2) where |61(z)| S M915
and |62(Z)| S Mozia for large values of z.

Let gf 1 denote the inverse branch of g that sends infinity to itself. In this setting

the statement we have to prove is: if I C (—00, —R] for some large R, then

|gf"I| x 1 and V3: 6 gnt [ml x n (5.0.2)

We will start by proving the second estimate. For z 6 (—00, —R] let gf1(z) = w. We
get '

w+1+E+61(w)=z=> w2+w(z—1)+A=w01(w)6[—1,1]

Solving the double inequality we get that

 

 

w< z—1—\/(Z—21)2_4(A_1)andw2 z—l—\/(z—21)2—4(A+1)

 

 

We will show that z_1_\/(Z_;)2_4(A—1) S z —% and z_1_\/(z‘1)2-4(A+1)

3
2 2 :1: — 5. These
inequalities are equivalent to :

4(A—1)—1S—2z and —2221—4(A+1)

which are true for z 6 (—00, —R]. Applying n times 2 — g S gl‘1(z) S 2 — % we get

it
2 9 _ 5 (5.0.3)

To get the first estimate of (5.0.2),
gl—nI : [_xn

letI —

[—$0 - 0.0, —$0] and
— an, -—:1:,,]. We have that:

g(—$n+1 _ an+1) : "xn _ an = g(_$n+l) _ an

80

 

 

 

 

 

 

 

 

+ 1 + A B 6 ( + a )
33,, an —- — —— cc 2
+1 +1 $n+1 + an+1 ($n+1 + an+1)2 3 "+1 "H
A B
= on + $n+1 — 1 + — 2 — 63($n+1)
$n+l xn+l
A B 22:” + an B’
z) an+1[1— — 2( +1 +l)2l Z a" _ 3

$n+1(-Tn+1 + an+l) $n+l($n+l + an+l) $n+l

Using the inequality (5.0.3) we get that
B” A(n) B"
an + 'n—3 Z an+1[1+ n2 ] 2 an — Til—3 (5.04)

for some positive constant B” and for some A(n) bounded away from zero and
infinity. Let 0,, = 021123 my where a small enough to guarantee a,- 2 fl; for
i = 1, . . . no. Notice that the product that defines on is bounded away from zero. We
will prove by induction that an 2 fin

Assume the inequality holds for n, then by (5.0.4)

71 n
an+121+An
"'2

It is enough to show that the last ratio is greater than %. This is equivalent

to:
i_._B:>; <=>
\ffl Cnn3—\/’n+1
1 Bl!

 

<2) > —
u(n +1)(\/T_l + x/n +1) “ cnn3
which is true for n 2 no if no was selected large enough.

Use an 2 3f; in the inequality (5.0.4) to get

coanB"
n25

 

A
a...[1+ #1 2 a.
n

81

therefore

A! 008”
an-l-l 2 an“ _ h3- _ ”25

 

].

This defines a convergent product, so inf an > 0.

From the first inequality of (5.0.4) we immediately get sup on < oo.

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[All

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