Conformal fractals, dimensions and ergodic theory

252 Pages·2000·1.433 MB·English

by  Przytycki F., Urbanski M.

11.11.1999
Chapters already renumbered, 2b=3 et
CONFORMAL FRACTALS, DIMENSIONS AND ERGODIC THEORY
Feliks Przyty
ki & Mariusz Urban(cid:19)ski
This book is an introdu
tion to the theory of iteration of non-uniformly expanding
holomorphi
maps and topi
s in geometri
measure theory of the underlying invariant
fra
tal sets. Probability measures on these sets yield informations on Hausdor(cid:11) and other
fra
tal dimensions and properties. The book starts with a
omprehensive
hapter on
abstra
t ergodi
theory followed by
hapters on uniform distan
e expanding maps and
thermodynami
al formalism. This material is appli
able in many bran
hes of dynami
al
systems and related (cid:12)elds, far beyond the appli
ations in this book.
Popular examples of the fra
tal sets to be investigated are Julia sets for rational
fun
tions on the Riemann sphere. The theory whi
h was initiated by Gaston Julia [J℄ and
Pierre Fatou [F℄ be
ame very popular sin
e the time when Benoit Mandelbrot's book [M℄
with beautiful
omputer made pi
tures appeared. Then it be
ame a (cid:12)eld of spe
ta
ular
a
hievements by top mathemati
ians during the last 20 years.
2
Consider for example the map f(z) = z for
omplex numbers z. Then the unit
1 1 1 (cid:0)1 1

ir
le S = fjzj = 1g is f-invariant, f(S ) = S = f (S ). For
(cid:25) 0;
=6 0 and
2
f
(z) = z +
, there still exists an f
-invariant set J(f
)
alled the Julia set of f
,
lose
1 1
to S , homeomorphi
to S via a homeomorphism h satisfying equality f Æ h = h Æ f
.
However J(f
) has a fra
tal shape. For large
the
urve J(f
) pin
hes at in(cid:12)nitely many
points; it maypin
h everywhere tobe
ome a dendrite, or even
rumble tobe
ome a Cantor
set.
These sets satisfy two mainproperties, standard attributes of "
onformalfra
tal sets":
1. Their fra
tal dimensions are stri
tly larger than the topologi
al dimension. 2. They
are
onformally "self-similar", namely arbitrarily small pie
es have shapes similar to large
pie
es via
onformal mappings, here via iteration of f.
To measure fra
tal sets invariant under holomorphi
mappings one applies probability
measures
orrespondingtoequilibriainthethermodynami
alformalism. Thisisabeautiful
example of interla
ing of ideas from mathemati
s and physi
s.
A prototype lemma [B, Lemma 1.1℄ at the roots of the thermodynami
al formalism
says that for given real numbers a1;:::;an the quantity
n n
X X
F(p1;:::pn) = (cid:0)pilogpi + pi(cid:30)i
i=1 i=1
Pn (cid:30)i
has maPximum value P = log i=1e as (p1;:::;pn) ranges over the simplex f(p1;:::;pn) :
n
pi (cid:21) 0; i=1pi = 1g and the maximum is assumed only at
n
(cid:30)j(cid:0)X (cid:30)i(cid:1)(cid:0)1
p^j = e e
i=1
1
We
an read (cid:30)i;pi;i = 1;:::;n as a fun
tion (potential), resp. probability distribution, on
the (cid:12)nite spa
e f1;:::;ng. Let us further follow Bowen [B℄: The quantity
n
X
S = (cid:0)pilogpi
i=1
is
alled entropy of the distribution (p1;:::;pn). The maximizing distribution (p^1;::;p^n) is

alled Gibbs or equilibrium state. In statisti
al me
hani
s (cid:30)i = (cid:0)(cid:12)Ei, where (cid:12) = 1=kT, T
is a temperature of an external "heat sour
e" and k a physi
al (Boltzmann)
onstant. The
P
n
quantity E = i=1piEi is the average energy. The Gibbs distribution maximizes then
the expression
1
S (cid:0)(cid:12)E = S (cid:0) E
kT
or equivalently minimizes the so-
alled free energy E (cid:0) kTS. The nature prefers states
with low energy and high entropy. It minimizes free energy.
TheideaofGibbsdistributionaslimitofdistributionson(cid:12)nitespa
esof
on(cid:12)gurations
of states (spins for example) of intera
ting parti
les over in
reasing growing to 1 bounded
d
parts of the latti
e ZZ introdu
ed in statisti
al me
hani
s (cid:12)rst by Bogolubov and Ha
et
[BH℄ and playing there a fundamental role was applied in dynami
al systems to study
Anosov (cid:13)ows and hyperboli
di(cid:11)eomorphisms at the end of sixties by Ja. Sinai, D. Ruelle
and R. Bowen. For more histori
al remarks see [Ru℄ or [Si℄. This theory met the notion of
entropyS borrowedfrominformationtheoryandintrodu
edbyKolmogorovasaninvariant
of a measure-theoreti
dynami
al system.
Later the usefulness of these notions to the geometri
dimensions has be
ome appar-
ent. It was present already in [Billingsley℄ but
ru
ial were papers by Bowen [Bo1℄ and
M
Cluskey & Manning [M
M℄.
In order to illustrate the idea
onsider the following example: Let Ti : I ! I, i =
1;:::;n > 1, where I = [0;1℄ is the unit interval, Ti(x) = (cid:21)ix + ai, where (cid:21)i;ai are real
numbers
hosen in su
h a way that all the sets Ti(I) are pairwise disjoint and
ontained
in I. De(cid:12)ne the limit set (cid:3) as follows
1
\ [ [
(cid:3) = Ti0 Æ:::ÆTik(I) = lim Ti0 Æ:::ÆTik;
k!1
k=0(i0;:::;ik) (i0;i1:::)
the latter union taken over all in(cid:12)nite sequen
es (i0;i1;:::), the previous over sequen
es of
length k +1.
It o

urs that its Hausdor(cid:11) dimension is equal to the only number (cid:11) for whi
h
(cid:11) (cid:11)
j(cid:21)1j +:::+j(cid:21)nj = 1:
(cid:3) is a Cantor set. It is self-similar with small pie
es similar to large pie
es with the use
(cid:0)1
of linear (more pre
isely, aÆne) maps (Ti0 Æ:::ÆTik) . We
a(cid:0)ll su
h a(cid:1)C(cid:11)antor set linear.
We
an distribute measure (cid:22) by setting (cid:22)(Ti0 Æ:::ÆTik(I)) = (cid:21)i0:::(cid:21)ik . Then for ea
h
interval J (cid:26) I
entered at a point of (cid:3) its diameter raised to the power (cid:11) is
omparable
2
to its measure (cid:22) (this is immediate for the intervals Ti0 Æ ::: Æ Tik(I)). (A measure with
this property for all small balls
entered at a
ompa
t set, in a eu
lidean spa
e of any
P
(cid:11)
dimension, is
alled a geometri
measure.) Hen
e (diamJ) is bounded away from 0 and
1 for all e
onomi
al (of multipli
ity not ex
eeding 2)
overs of (cid:3) by intervals J.
Note that for ea
h k (cid:22) restri
ted to the spa
e of unions of Ti0 Æ ::: Æ Tik(I), ea
h
su
h interval viewed as one point, is the Gibbs distribution, where we set (cid:30)((i0;:::;ik)) =
P
(cid:30)(cid:11)((i0;:::;ik)) = l=0;:::;k(cid:11)log(cid:21)il. The number (cid:11) is the unique 0 of the pressure fun
tion
P
1 (cid:30)a((i0;:::;ik))
P((cid:11)) = log e . In this spe
ial aÆne example this is independent
k+1 (i0;:::;ik)
of k. In general non-linear
ase to de(cid:12)ne pressure one passes with k to 1.
The family Ti and
ompositions is an example of very popular in re
ent years Iterated
Fun
tion System [Barnsley℄. Note that on a neighbourhood of ea
h Ti(I) we
an
onsider
^ (cid:0)1 ^
T := T . Then (cid:3) is an invariant repeller for the distan
e expanding map T.)
i
The relations between dynami
s, dimension and geometri
measure theory start in
our book with the theorem that the Hausdor(cid:11) dimension of an expanding repeller is the
1+"
unique 0 of the adequate pressure fun
tion for sets built with the help of C usually
d
non-linear maps in IR or
onformal maps in IR .
This theory was developed for non-uniformly hyperboli
maps or (cid:13)ows in the setting
of smooth ergodi
theory, see [HK℄, by Man~(cid:19)e [M℄, Lai-Sang-Young and Ledrappier [LY℄;
see [Pesin℄ for re
ent developments. The advan
ed
hapters of our book are devoted to this
theory, but we restri
t ourselves to
omplex dimension 1. So the maps are non-uniformly
expanding and the main te
hni
al diÆ
ulties are
aused by
riti
al points, where we have
strong
ontra
tion sin
e the derivative by de(cid:12)nition is equal to 0 at
riti
al points.
A dire
tion not developed in this book are Conformal Iterated Fun
tion Systems with
in(cid:12)nitely many generators Ti. They o

ur naturally as return maps in many important

onstru
tions, for example for rational maps with paraboli
periodi
points or in the In-
du
ed Expansion
onstru
tion for polynomials [GS℄. Beautiful examples are provided by
in(cid:12)nitely generated Kleinian groups [.℄. The systemati
treatment of Iterated Fun
tion
Systems with in(cid:12)nitely many generators
an be found in [MU1℄, [MU2℄, [MU3℄, [MPU℄
and [U1℄ for example.
Below is a short des
ription of the
ontent of the book.
Chapter 1 is an introdu
tion to abstra
t ergodi
theory, here T is a probability mea-
sure preserving transformation. The reader will (cid:12)nd proofs of the fundamental theorems:
Birkho(cid:11) Ergodi
Theorem and Shannon-M
Millan-Breiman Theorem. We introdu
e en-
tropy, measurable partitions and dis
uss
anoni
al systems of
onditional measures in
Rohlin's Lebesgue spa
e the notion of natural extension (inverse limit in the appropriate

ategory). We follow here Rohlin's Theory [Ro℄, see also [FKS℄. Next to prepare to ap-
pli
ations for (cid:12)nite-to-one rational maps we sket
h Rohlin's theory on
ountable-to-one
endomorphisms and introdu
e the notion of Ja
obian, see also [Parry℄. Finally we dis
uss
mixing properties (K-propery, exa
tness, Bernoulli) and probability laws (Central Limit
Theorem, abbr. CLT, Law of Iterated Logarithm, LIL, Almost Sure Invarian
e Prin-

iple, ASIP) for the sequen
e of fun
tions (random variables on our probability spa
e)
n
(cid:30)ÆT ;n = 0;1;:::.
Chapter 2 is devoted to ergodi
theory and termodynami
al formalism for general
3
ontinuousmapson
ompa
tmetri
spa
es. Themainpointhereistheso
alledVariational
Prin
iple for pressure,
ompare the prototype lemma above. We apply also fun
tional
analysis in order to explain Legendre transform duality between entropy and pressure.
We follow here [Israel℄ and [Ruelle℄. This material is appli
able in large deviations and
multifra
tal analysis, and is dire
tly related to the uniqueness of Gibbs states question.
In Chapters 1, 2 we often follow the beautiful book by Peter Walters [Wa℄.
In Ch 3. distan
e expanding maps are introdu
ed. Analogouslyto AxiomA di(cid:11)eomor-
phisms [Smale, Bowen℄ or endomorphisms [Przy℄ we outline a topologi
al theory: spe
tral
de
omposition, spe
i(cid:12)
ation, Markov partition, and start a"bounded distortion" playwith
Ho(cid:127)lder
ontinuous fun
tions.
In Chapter 4 termodynami
al formalism and mixing properties of Gibbs measures for
open distan
e expanding maps T and H(cid:127)older
ontinuous potentials (cid:30) are studied. To large
extend we follow [Bo℄ and [Ru℄. We prove the existen
e of Gibbs measures (states): m
with Ja
obian being exp(cid:0)(cid:30) up to a
onstant fa
tor, and T-invariant (cid:22) = (cid:22)(cid:30) equivalent
P
to m. The idea is to use the transfer operator L(cid:30)(u)(x) = y2T(cid:0)1(x)u(y)exp(cid:30)(y) on the
Bana
h spa
e of Ho(cid:127)lder
ontinuous fun
tions u. We prove the exponential
onvergen
e
R
(cid:0)n n
(cid:24) L(cid:30)(u) ! ( udm)u(cid:30), where (cid:24) is the eigenvalue of the largest absolute value and u(cid:30)
the
orresponding eigenfun
tion. One obtains u(cid:30) = dm=d(cid:22). We dedu
e CLT, LIL and
ASIP, and the Bernoulli property for the natural extension.
We provide three di(cid:11)erent proofs of the uniqueness of the invariant Gibbs measure.
The (cid:12)rst, simplest, follows [Keller???℄, the se
ond relies on the prototype lemma, the third
one on the di(cid:11)erentiability of the pressure fun
tion in adequate fun
tion dire
tions.
Finally we prove Ruelle's formula
Z n(cid:0)1 Z n(cid:0)1 Z
X X
1
2 i i
d P((cid:30)+tu+sv)=dtdsjt=s=0 = lim ( (uÆT (cid:0) ud(cid:22)(cid:30))(cid:1)( (vÆT (cid:0) vd(cid:22)(cid:30))d(cid:22)(cid:30):
n!1 n
i=0 i=0
2 n
This expression for u = v is equal to (cid:27) in CLT for the sequen
e uÆT and measure (cid:22)(cid:30).
(In the book we use the letter T to denote a measure preserving transformation.
Maps preserving an additional stru
ture,
ontinuous smooth or holomorphi
for example,
are usually denoted f or g.)
In Chapter 5 the metri
spa
e with the a
tion of an open distan
e expanding map
is embedded in a smooth manifold and it is assumed that the map smoothly extends
to a neighbourhood. We
all the spa
e with the extended dynami
s: Smooth Expanding
Repeller, abbr. SER. We study smoothness of the density u(cid:30). Finally we provide in
detail D. Sullivan's theory
lassifying line Cantor sets via s
aling fun
tion, sket
hed in [Su℄
and dis
uss the realization problem [PT℄. We also dis
uss appli
ations for solenoids for
Feigenbaum maps.
In Chapter 6 we provide de(cid:12)nitions of various "fra
tal dimensions": Hausdor(cid:11), box
and pa
king. We
onsider also Hausdor(cid:11) measures with gauge fun
tions difefrent from
(cid:11)
t . We prove "Volume Lemma" linking, roughly speaking, (global) dimension with lo
al
dimensions.
In Chapter 7 we (cid:12)nally introdu
e Conformal Expanding Repellers, abbr. CER, and
?(cid:0)1
relate pressure with Hausdor(cid:11) dimension. We prove C dependen
e of the dimension on
4
?
the parameter if the dependen
e on the parameter of the expanding map is C . We deal
with smooth repellers in IR and
onformal repellers in CI. Here 2 <? (cid:20) !, the real analyti

ase.
Next we follow the easy (uniform) part of [PUZ℄. We prove that for CER (X;f) and
H(cid:127)older
ontinuous (cid:30) : X ! R, for (cid:20) = HD((cid:22)(cid:30)), Hausdor(cid:11) dimension of the Gibbs measure
(cid:22)(cid:30) (in(cid:12)mum of Hausdor(cid:11) dimensions of sets of full measure), either HD(X) = (cid:20) the
measure (cid:22)(cid:30) is equivalent to (cid:3)(cid:20), the Hausdor(cid:11) measure in dimension (cid:20), and is a geometri
measure, or(cid:22)(cid:30) is singular withrespe
t to(cid:3)(cid:20) and theright gaugefun
tion for the Hausdor(cid:11)
p
(cid:20)
measure to be
ompared to (cid:22)(cid:30) is (cid:8)((cid:20)) = t exp(
log1=tlogloglog1=t). In the proof we
use LIL. This theorem is used to prove a di
hotomy for the harmoni
measure on a Jordan

urve , bounding a domain (cid:10), whi
h is a repeller for a
onformal expanding map. Either
 is real analyti
or harmoni
measure is
omparable to the Hausdor(cid:11) measure with gauge
fun
tion (cid:8)(1). This yields an information about the lower and upper growth rates of
0
jR (r(cid:16))j, for r % 1, for almost every (cid:16) with j(cid:16)j = 1 and univalent fun
tion R from the
unit dis
jzj < 1 to (cid:10). This is a dynami
al
ounterpart of Makarov's theory of boundary
behaviour for general simply
onne
ted domains, [Makarov℄.
2
We prove in parti
ular that for f
(z) = z +
;
6= 0;
(cid:25) 0 1 < HD(J(f
)) < 2.
We show how to express in the language of pressure another interesting fun
tion:
R
0 t
jR (r(cid:16))j jd(cid:16)j for r % 1.
j(cid:16)j=1
We also look
loser at the Gibbs measures, dis
uss so
alled multifra
tal analysis, and
study large deviations.
Finally we apply our theory to the boundary of von Ko
h "snow(cid:13)ake" and more
general Carleson fra
tals.
Chapter 8 is devoted to Sullivan's rigidity theorem, saying that two non-linear ex-
panding repellers (X;f);(Y;g)that are Lips
hitz
onjugate (or more generally there exists
a measurable
onjuga
y that transforms a geometri
measure on X to a geometri
measure
on Y, then the
onjuga
y extends to a
onformal one. This means that measures
lassify
non-linear
onformal repellers. This fa
t, annou
ed in [Su℄ only with a sket
h of the proof,
is proved here rigorously for the (cid:12)rst time. We sket
h also a generalization by E. Prado.
In Chapter 9 we start to deal with non-uniform expanding phenomena. A heart of
this
hapter is the proof of the formula HD((cid:22)) = h(cid:22)(f)=(cid:31)R(cid:22)(f) for an arbitrary f-invariant
0
ergodi
measure (cid:22) of positive Laypunov exponent (cid:31)(cid:22) := logjf jd(cid:22).
(The word non-uniform expanding is used just to say that we
onsider (typi
al points
of) an ergodi
measure with positive Lyapunov exponent. In higher dimension one uses
the name non-uniform hyperboli
for measures with all Lyapunov exponents non-zero.)
It is so roughly be
ause a small dis
around z, whose n-th image is large, has diameter
n 0 (cid:0)1
of order j(f ) (z)j (cid:25) exp(cid:0)n(cid:31)(cid:22) and measure exp(cid:0)nh(cid:22)(f) (Shannon-M
Millan-Breiman
theorem is involved here)
Chapter 10 is devoted to
onformal measures, namely probability measures with Ja-
0 (cid:11)

obian Constexp(cid:0)(cid:30) or more spe
i(cid:12)
ally jf j in a non-uniformly expanding situation, in
parti
ular for any rational mapping f on its Julia set J. It is proved that there exists a
minimal exponent Æ(f) for whi
h su
h a measure exists and that Æ(f) is equal to ea
h of
the following quantities:
Dynami
al Dimension DD(J) := supfHD((cid:22))g, where (cid:22) ranges over all ergodi
f-
5
invariant measures on J of positive Lyapunov exponent.
Hyperboli
Dimension HyD(J) := supfHD(Y)g, where Y ranges over all Conformal
Expanding Repellers in J, or CER's that are Cantor sets.
It is an open problem whether for every rational mapping HyD(J) = HD(J) = box
dimension of J, but for many nonuniformly expandig mappings these equalities hold. It
is often easier to study the
ontinuity of Æ(f) with respe
t to a parameter, than dire
tly
Hausdor(cid:11) dimension. So one obtains an information about the
ontinuity of dimensions
due to the above equalities.
Most of the book was written in the years 1990-1992 and was le
tured to graduate
students by ea
h of us in Warsaw, Yale and Denton. We negle
ted (cid:12)nishing writing, but
re
ently unexpe
tedly to us the methods in Chapter 10, relating hyperboli
dimension to
minimal exponent of
onformal measure, were used to study the dependen
e on " of the
2
dimension of Julia set for z +1=4+", for " ! 0 and other paraboli
bifur
ations, by A.
Douady, P. Sentena
and M. Zinsmeister in [DSZ℄ and by C. M
Mullen in [M
M℄. So we
de
ided to make a (cid:12)nal e(cid:11)ort. Meanwhile ni
e books appeared on some topi
s of our book,
let us mention [Fal
oner℄, [Zinsmeister℄, [Gora,Boyarsky℄, [Viana℄, but a lot of important
material in our book is new or was hardly a

essible, or is written in an un
onventional
way.
[Barnsley℄ ..................
[Fal
oner℄ K. Fal
oner, Te
hni
s in Fra
tal Geometry
[Zinsmeister℄ M. Zinsmeister, Le Formalisme Thermodynamique: Mode d'emploi
[Boyarsky, G(cid:19)ora℄ A. Boyarsky, P. G(cid:19)ora, Laws of Chaos, Invariant Measures and Dy-
nami
al Systems in One Dimension. Birkh(cid:127)auser, Boston 1997
[Viana℄ M. Viana,
6
November 14, 1999

hapters renumbered, 2b=3 et
CHAPTER 0.
BASIC EXAMPLES AND DEFINITIONS
Let us start with de(cid:12)nitions of dimensions. We shall
ome ba
k to them in a more
systemati
way in Chapter 5.
De(cid:12)nition 0.1. Let (X;(cid:26)) be a metri
spa
e. We
all upper (lower) box dimension
of X the quantity
logN(r)
limsup(liminf)r!0
(cid:0)logr
where N(r) is the minimal number of balls of radius r whi
h
over X.
Sometimes the names
apa
ity or Minkowski dimension or box-
ounting dimension are
used. The nameboxdimension
omes fromthe situationwhere X is a subset of aeu
lidean
d (cid:0)n (cid:0)n
spa
e IR . Then one
an
onsider only r = 2 and N(2 )
an be repla
ed by the number
k1 k1+1 kd kd+1
of dyadi
boxes [2(cid:0)n; 2(cid:0)n ℄(cid:2):::(cid:2)[2(cid:0)n; 2(cid:0)n ℄;kj 2 ZZ interse
ting X.
De(cid:12)nition 0.2. Let (X;(cid:26)) be a metri
spa
e. For every (cid:20) > 0 we de(cid:12)ne (cid:3)(cid:20)(X) =
P1 (cid:20)
limÆ!0inff i=1(diamUi) g, where the in(cid:12)mum is taken over all
ountable
overs (Ui;i =
1;2;:::) of X by sets of diameter not ex
eeding Æ. (cid:3)(cid:20)(Y) de(cid:12)ned as above on all subsets
Y (cid:26) X is
alled (cid:20)-th outer Hausdor(cid:11) measure.
It is easy to see that there exists (cid:20)0 : 0 (cid:20) (cid:20)0 (cid:20) 1 su
h that for all (cid:20) : 0 (cid:20) (cid:20) < (cid:20)0
(cid:3)(cid:20)(X) = 1 and for all (cid:20) : (cid:20)0 < (cid:20) (cid:3)(cid:20)(X) = 0. The number (cid:20)0 is
alled the Hausdor(cid:11)
dimension of X.
Notethatifinthisde(cid:12)nitionwerepla
etheassumption: setsofdiameternotex
eeding
Æ by equal Æ, and limÆ!0 by liminf or limsup, we obtain box dimension.
A standard example to
ompare both notions is the set f1=n;n = 1;2;:::g in IR. Its
(cid:0)n
box dimension is equal to 1/2 and Hausdor(cid:11) dimension is 0. If one
onsiders f2 g instead
one obtains both dimensions 0. Also linear Cantor sets in Introdu
tion have Hausdor(cid:11) and
box dimensions equal. The reason for this is self-similarity.
d
Example 0.3. Shifts spa
es. For every natural number d
onsider the spa
e (cid:6) of
all in(cid:12)nite sequen
es (i0;i1;:::) with in 2 f1;2;:::;dg. Consider the metri
1
X
0 0 n 0
(cid:26)((i0;i1;:::);(i0;i1;:::))= (cid:21) jin (cid:0)inj
n=0
for an arbitrary 0 < (cid:21) < 1. Sometimes it is more
omfortable to use the metri
0
(cid:26)((i0;i1;:::);(i00;i01;:::))= (cid:21)(cid:0)minfn:in6=ing
d d
, equivalent to the previous one. Consider (cid:27) : (cid:6) ! (cid:6) de(cid:12)ned by f((i0;i1;:::) = (i1;:::).
d
The metri
spa
e ((cid:6) ;(cid:26)) is
alled one-sided shift spa
e and the map (cid:27) the left shift. Often,
1
if we do not spe
ify metri
but are interested only in the
artesian produ
t topology in
+
d ZZ
(cid:6) = f1;:::;dg , we use the name topologi
al shift spa
e.
~d
One
an
onsider the spa
e (cid:6) of all two sides in(cid:12)nite sequen
es (:::;i(cid:0)1;i0;i1;:::).
This is
alled two-sided shift spa
e.
d
Ea
h point (i0;i1;:::) 2 (cid:6) determines its forward traje
tory under (cid:27), but is equipped
with a Cantor set of ba
kward traje
tories. Together with the topology determined by
P
1 jnj 0 ~d
the metri
n=(cid:0)1(cid:21) jin (cid:0)inj the set (cid:6)
an be identi(cid:12)ed with the inverse limit (in the
d d
topologi
al
ategory) of the system ::: ! (cid:6) ! (cid:6) where all the maps ! are (cid:27).
Note that the limit Cantor set (cid:3) in Introdu
tion, with all (cid:21)i = (cid:21) is Lips
hitz homeo-
T
d
morphi
to (cid:6) , with the homeomorphism h mapping (i0;i1;:::) to kTi0 Æ::ÆTik(I). Note
(cid:0)1
that for ea
h x 2 (cid:3), h (x) is the sequen
e of integers (i0;i1;:::) su
h that for ea
h k,
^k
T (x) 2 Tik(I). Itis
alleda
oding sequen
e. If weallowtheendpoinPtsofTi(I)tooverlap,
(cid:0)1 1 (cid:0)k(cid:0)1
in parti
ular (cid:21) = 1=d and ai = (i(cid:0)1)=d, then (cid:3) = I and h (x) = k=0(ik (cid:0)1)d .
One generalizes the one (or two) -sided shift spa
e,
alled sometimes full shift spa
e
by
onsidering the set (cid:6)A for an arbitrary d(cid:2)d { matrixA = (aij with aij = 0 or 1 de(cid:12)ned
by
d
(cid:6)A = f(i0;i1;:::) 2 (cid:6) : aitit+1 = 1 for every t = 0;1;:::g:
By the de(cid:12)nition (cid:27)((cid:6)A (cid:26) (cid:6)A. (cid:6)A with the mapping (cid:27) is
alled a topologi
al Markov

hain. Here the word topologi
al is substantial, otherwise it is
ustomary to think of a
(cid:12)nite number of states sto
hasti
pro
ess, see Example 0.8.
Example 0.4. Iteration of rational maps. Let f : CI ! CI be a holomorphi
mapping of the Riemann sphere CI. Then it must be rational, i.e. ratio of two polynomials.
We assume that the topologi
al degree of f is at least 2. The Julia set J(f) is de(cid:12)ned as
follows:
n
J(f) = fz 2 CI : 8U 3 z ;U open, the family of iterates f = f Æ:::ÆfjU, n times, for
n = 1;2;::: is not normal in the sense of Montel g.
A familyof holomorphi
fun
tions ft : U ! CI is
alled normal (in the sense of Montel)
if it is pre-
ompa
t, namely from every sequen
e of fun
tions belonging to the family one

an
hoose a subsequen
e uniformly
onvergent (in the spheri
al metri
on the Riemann
sphere CI) on all
ompa
t subsets of U.
n
z 2 J(f) implies for example, that for every U 3 z the family f (U)
overs all CI but
n
at most 2 points. Otherwise by Montel's theorem ff g would be normal on U.
Another
hara
terizationofJ(f)isthatJ(f)isthe
losureofrepellingperiodi
points,
n
namely those points z 2 CIfor whi
h there exists an integer n su
h that f (z) = z and
n 0
j(f ) (z)j > 1.
n 0
There is only (cid:12)nite number of attra
ting periodi
points, j(f ) (z)j < 1; they lie
outside J(f), an un
ountable "
haoti
, repelling" Julia set. The la
k of symmetry between
atra
ting and repelling phenomena is
aused by the non-invertibility of f.
It is easy to prove that J(f) is
ompa
t,
ompletely invariant: f(J(f)) = J(f) =
(cid:0)1
f (J(f)), either nowhere dense or equal to the whole sphere (to prove this use Montel's
theorem).
n
For polynomials, the set of points whose images under iterates f ;n = 1;2;:::, tend
to 1, basin of attra
tion to 1, is
onne
ted and
ompletely invariant. Its boundary is the
2
Julia set.
Che
k that all these general de(cid:12)nitions and statements are
ompatible with the dis-
2

ussion of f(z) = z +
in Introdu
tion. As introdu
tion to this theory we re
ommend for
example the books [Beardon℄, [Carleson, Gamelin℄ and [Steinmetz℄.
Below are the
omputer pi
tures of some Julia sets
FIGURES: Rabbit, Sierpinski
arpet(rationalfun
tion of degree 2), Newton's method
A Julia set
an have Hausdor(cid:11) dimension arbitrarily
lose to 0 (but not 0) and ar-
bitrarily
lose to 2 and even exa
tly 2 (being in the same time nowhere dense). It is not
known whether it
an have positive Lebesgue measure. We shall
ome ba
k to these topi
s
in Chapters 6, 10.
Example 0.5. Complex linear fra
tals. The linear Cantor set
onstru
tion in IR
des
ribed in Introdu
tion
an be generalized to
onformal linear Cantor and other fra
tal
sets in CI:
Let U (cid:26) CI be a bounded
onne
ted domain and Ti(z) = (cid:21)iz + ai, where (cid:21)i;ai are

omplex numbers, i = 1;:::;n> 1. Assume that
losures
lTi(U) are pairwise disjoint and

ontained in U. The limit Cantor set (cid:3) is de(cid:12)ned in the same way as in Introdu
tion.
In Ch.7 we shall prove that it
annot be the Julia set for a holomorphi
extension of
^ (cid:0)1
T = Ti on Ti(U) for ea
h i, to the whole sphere CI.
If we allow that the boundaries of Ti(U) interse
t or interse
t U we obtain other
interesting examples
FIGURES: Sierpinski
arpet, Sierpinski gasket, boundary of von Ko
h snow(cid:13)ake
Examples 0.6. A
tion of Kleinian groups. Beautiful examples of fra
tal sets
arise as limit sets of the a
tion of Kleinian groups on CI.
LetHobethegroupofallhomographies,namelytherationalmappingsoftheRiemann
az+b
sphere of degree 1, i.e. of the form z 7! where ad(cid:0)b
6= 0. Every dis
rete subgroup

z+d
of Ho is
alled Kleinian group. If all the elements of a Kleinian group preserve the unit
dis
ID = fjzj < 1g, the group is
alled Fu
hsian.
Consider for example a regular hyperboli
4n-gon in ID (equipped with the hyperboli
j
metri
)
entered at 0. Denote the
onse
utive sides by a ;i = 1;:::;n;j = 1;:::;4 in the
i
1 4 1 j
lexi
ographi
al order: a ;:::a ;a ;:::. Ea
h side is
ontained in the
orresponding
ir
le C
1 1 2 i
j j
interse
ting ID at the right angles. Denote the dis
bounded by C by D .
i i
j j+2
It is not hard to see that the
losures of D and D are disjoint for ea
h i and
i i
j = 1;2.
FIGURE: regular hyperboli
o
tagon, ID=G.
j j j+2 j
Letg ;j = 1;2betheuniquehomographypreserving ID mappinga toa and D to
i i i i
j+2 j
the
omplementof
lD . Itis easy tosee thatthefamilyfg ggenerates aFu
hsian group
i i S
G. For an arbitrary Kleinian group G, the Poin
ar(cid:19)e limit set (cid:3)(G) = limk!1gk(z),
3
the union taken over all sequen
es of pairwise di(cid:11)erent gk 2 G su
h that gk(z)
onverges,
where z is an arbitrary point in CI. It is not hard to prove that (cid:3)(G) does not depend on
z.
j j
For the above example (cid:3)(G) = ID. If we
hange slightly g (the
ir
les C
hange
i i
j
slightly), then either (cid:3)(G) is a
ir
le S (all new C interse
t S at the right angle), or it
i
2
is a fra
tal Jordan
urve. The phenomenon is similar to the
ase of the maps z 7! z +
des
ribed in Introdu
tion. For details see [Bowen℄, [Bowen, Series℄, [Sullivan℄. We provide
a sket
h of the proof in Chapter 7.
j
If all the
losures of the dis
s D ;i = 1;:::;n;j = 1;:::;4 be
ome pairwise disjoint,
i
(cid:3)(G) be
omes a Cantor set (the group is
alled then a S
hottky group or a Kleinian group
of S
hottky type).
Examples 0.7. Higher dimensions. Though the book is devoted to 1-dimensional
real and
omplex iteration and arising fra
tals, Chapters 1-3 apply to general situations.
2
A basi
example is Smale's horseshoe. Take a square K = [0;1℄(cid:2) [0;1℄ in the plane IR
and map it aÆnely to a strip by squeezing in the horizontal dire
tion and stret
hing in
1 1 1
the verti
al, for example f(x;y) = ( x+ ;3y(cid:0) ) and bend the strip by a new map g so
3 9 3
1 4 4 8 5 8 1
that the re
tangle [ ; ℄(cid:2)[ ; ℄ is mapped to [ ; ℄(cid:2)[(cid:0) ;1℄. The resulting
omposition
9 9 3 3 9 9 3
T = gÆf maps K to a "horseshoe", see [Smale, p.773℄
FIGURE: horseshoe, stadium extension
1
The map
anbeeasilyextendedtoaC -di(cid:11)eomorphismofCI bymappinga"stadium"
extending K to a bent "stadium", and its
omplement to the respe
tive
omplement. The
K n
set (cid:3) of points not leaving K under a
tion of T ;n = :::;(cid:0)1;0;1;:::is the
artesian prod-
u
t of two Cantor sets. This set is T-invariant, "uniformly hyperboli
". In the horizontal
dire
tion we have
ontra
tion, in the verti
al dire
tion uniform expansion. The situation
d
is di(cid:11)erent from the previous examples of (cid:6) or linear Cantor sets, where we had uniform
expansion in all dire
tions.
Smale's horseshoe is a universal phenomenon. It is always present for an iterate of a
di(cid:11)eomorphism f having a transversal homo
lini
point q for a saddle p. The stable and
s n u (cid:0)n
unstable manifolds W (p) := fy : f (y) ! pg;W (p) := fy : f (y) ! pg as n ! 1,
interse
t transversally at q. For more details on hyperboli
sets see [HK℄.
FIGURE: homo
lini
point and embedded horseshoe.
Note that Tj(cid:3)K is topologi
ally
onjugate to the left shift (cid:27) on the two-sided shift
~2 K ~2
spa
e (cid:6) , namely there exists a homeomorphism h : (cid:3) ! (cid:6) su
h that h Æ T = (cid:27) Æ h.
K ^
Compare h i Example 0.3. T on (cid:3) is the inverse limit of the mapping T on the Cantor
~2 2
set des
ribed in Introdu
tion, similarly to the inverse limit (cid:6) of (cid:27) on (cid:6) . The philosophy
is that hyperboli
systems appear as inverse limits of expanding systems.
s
A partition of a hyperboli
set (cid:3) into lo
al stable (unstable) manifolds: W (x) =
n n
fy 2 (cid:3) : (8n (cid:21) 0)(cid:26)(f (x);f (y)) (cid:20) "(x)g for a small positive measurable fun
tion ", is
4