przytycki f., urbanski m. conformal fractals, dimensions and ergodic theory

Chapters already renumbered, 2b=3CONFORMAL FRACTALS,DIMENSIONS ANDERGODIC THEORY Feliks Przyt ki& Mariusz Urbanski This book is an intro to the theory of iteration of non-uniformly expan

Chapters already renumbered, 2b=3

CONFORMAL FRACTALS,DIMENSIONS ANDERGODIC THEORY

Feliks Przyt ki& Mariusz Urbanski

This book is an intro to the theory of iteration of non-uniformly expanding

maps and in measure theory of the underlying invariant

sets Probabilitymeasureson these setsyield informations on Hausdor andother

dimensions and properties The book starts with a e hapter on

ergo theory followed by hapters on uniform expanding maps and

thermo al formalism This material is in many hes of

systems andrelated elds, far beyond the inthis book

Popular examples of the sets to be investigated are Julia sets for rational

onthe Riemannsphere The theory hwasinitiated b Gaston Julia[J℄ and

Pierre Fatou [F℄ b very popular the time whenBenoit Mandelbrot's book [M℄

with beautiful made appeared Then it b a eld of sp

hievements b top during the last 20 years

Consider for example the map f(z) = z

= f1

(S1

via a homeomorphism h satisfying equality f Æh = hÆf

However J(f ) has a shape For large the e J(f ) hes at in nitely many

points;itmay heverywheretob adendrite,oreven bletob aCantor

set

Thesesetssatisfytwomainproperties,standardattributesof sets":

1 Their dimensions are larger than the top dimension 2 They

are "self-similar",namelyarbitrarilysmall have shapes similar tolarge

via mappings, here via iterationof f

Tomeasure setsinvariantunder mappingsoneappliesprobability

measures ondingtoequilibriainthethermo formalism Thisisabeautiful

example of in of ideas from and ph

A prototype lemma [B, Lemma 1.1℄ at the roots of the thermo formalism

says that for given real numbers a

1

;:::;a

nthe quantity

F(p

1

;:::p

n)=n

X

i=1p

ilogp

i+n

X

i=1p

X

i=1e

i

1

the nite f1;:::;ng Let us further followBowen [B℄: The quantity

S =n

X

i=1p

ilogp

isatemperatureofan external"heat andk aph (Boltzmann) t The

quantity E =

P

n

i=1p

iE

or equivalently minimizes the fre energy E kTS The nature prefers states

with low energy and high entropy It minimizesfree energy

TheideaofGibbsdistributionaslimitofdistributionson nite of

of states(spinsforexample) ofin over growingto1bounded

parts of the Z

d

[BH℄ and playing there a fundamental role was applied in systems to study

Anosov ws and hyperb dieomorphisms atthe end of sixtiesb Ja Sinai, D.Ruelle

andR Bowen Formore remarkssee[Ru℄ or[Si℄ This theorymet thenotion of

entropyS borrowedfrominformationtheoryandintro byKolmogorovasaninvariant

Later the usefulness of these notions to the dimensions has b

appar-ent It was present already in [Billingsley℄ but were papers by Bowen [Bo1℄ and

ey & Manning

In order to illustrate the idea the following example: Let T

i: I ! I, i =

1;:::;n > 1, where I = [0;1℄ is the unit interval, T

i(x) = 

i

x+a

iwhere 

ia

iare real

numbers hosen in h a way that all the sets T

i(I) are pairwise disjoint and tained

in I De ne the limitset  as follows

=1

\

k=0[

(i

0

;:::;i

k)T

k!1T

+:::+j

nj

=1:

 is a Cantor set It is self-similar with small similar to large with the use

of linear (more , aÆne) maps (T

We h a Cantor set linear

We distribute measure  bysetting (T

i

0:::

this property for all small balls tered at a set, in a of any

dimension,is age measure.)

P

(diamJ)

isboundedawayfrom0and

1 for all (of m ynot 2) vers of  b intervals J

Note that for h k  to the of unions of T

h interval viewed as one point, is the Gibbs distribution, where we set ((i

0

;:::;i

k))=

a((i

0

;:::;i

k))

In this sp aÆne example this is independent

of k In generalnon-linear to de ne pressure one passes with k to 1

ThefamilyT

iand ositionsisanexample of very popularin tyears Iterated

F System [Barnsley℄ Note that on a neighbourhood of h T

i(I) we

The relations between dimension and measure theory start in

our book with the theorem that the Hausdor dimension of an expanding repeller is the

unique 0 of the adequate pressure for sets built with the help of C

This theory was developed for non-uniformly hyperb maps or ws in the setting

of smooth ergo theory, see [HK℄, b Ma~ne [M℄, Lai-Sang-Young and Ledrappier [LY℄;

see[Pesin℄for tdevelopments Theadv haptersof ourbookaredevotedtothis

theory, but we ourselves to dimension 1 So the maps are non-uniformly

strong the derivative byde nition is equal to 0 at points

A not developedinthisbookareConformal IteratedF Systemswith

in nitely many generators T

iThey o naturally as return maps in many important

for example for rational maps with parab perio points or in the

In-ed Expansion for polynomials [GS℄ Beautiful examples are provided by

in ... remarkssee[Ru℄ or[Si℄ This theorymet thenotion of

entropyS borrowedfrominformationtheoryandintro byKolmogorovasaninvariant

Later the usefulness of these notions to the dimensions has b

appar-ent... transform duality between entropy and pressure

We follow here [Israel℄ and [Ruelle℄ This material is in large deviations and

multifr analysis, and is related to the uniqueness of... Chapter4termo formalismandmixingproperties of Gibbsmeasures for

open expandingmapsT andHolder tinuouspotentialsarestudied Tolarge

extend we follow [Bo℄ and [Ru℄ We prove the of