On the variational principle for the topological entropy of certain non-compact sets

On the variational principle for the topological entropy of

certain non-compact sets

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Takens, F., & Verbitskiy, E. A. (2001). On the variational principle for the topological entropy of certain non-compact sets. (Report Eurandom; Vol. 2001007). Eurandom.

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On the variational principle for the

topological entropy of certain

non-compact sets

Floris Takensyand Evgeny Verbitskiyz

yDepartment of Mathematics, University of Groningen P.O.Box 800, 9700 AV,

Groningen, The Netherlands, (e-mail: f.takens@math.rug.nl)

zEurandom, Technical University of Eindhoven, P.O.Box 513, 5600 MB,

Eindhoven, The Netherlands, (e-mail: e.verbitskiy@tue.nl) December 1999, revised February 2001

Abstract

For a continuous transformationf of a compact metric space (Xd) and any continuous function'onX we consider sets of the form

K = n x2X: lim n!1 1 n n;1 X i=0 '(f i( x)) = o  2R:

For transformations satisfying the specication property we prove the following Variational Principle

htop(fK ) = sup  h(f) :is invariant and Z 'd=  

where htop(f) is the topological entropy of non-compact sets. Using this result we are able to obtain a complete decription of the multifractal spectrum for Lyapunov exponents of the so-called Manneville{Pomeau map, which is an interval map with an indierent xed point.

1 Introduction

Often the problems of multifractal analysis of local (or pointwise) dimensions and entropies are reduced to consideration of the sets of the following form

K =n x2X : lim n!1 1 n n;1 X i=0 '(fi_{(x)) =}o  2R

wheref :X !X is some transformation, and':X !R is a function, sometimes

called observable. Typically, f is a continuous transformation of some compact metric space (Xd) and'is suciently smooth.

In particular, one is interested in the \size" of these sets K . The following characteristics of the setsK have been studied in the literature:

D'() = dimH(K ) E'() =htop(fK )

where dimH(K ) andhtop(fK ) are the Hausdor dimension and the topological

entropy ofK , respectively. The precise denition of the topological entropy of non-compact sets will be given in section 3, but for now the topological entropy should be viewed as a dimension-like characteristic, similar to the Hausdor dimension. The functionsD'(),E'() will be called the dimension and entropy multifractal

spectra of'.

Recently similar problems were considered in the relation with a denition of a rotational entropy 8, 10].

Multifractal analysis studies various properties of the multifractal spectraD'(), E'() as functions of, e.g., smoothness and convexity, and relates these spectra to

other characteristics of a dynamical system. In order to obtain non-trivial results one typically has to make 2 types of assumptions: rstly, on the dynamical system (Xf), and secondly, on the properties of the observable function'. For example,

 (15], see also 16]) iff is a suciently smooth expanding conformal map, and

'is a Holder continuous function, thenE'() is real-analytic and convex.  (21]) if f is an expansive homeomorphism with specication, and ' has

bounded variation, thenE'() isC

1 and convex.

In both cases,E'() is a Legendre transform of a pressure functionP'(q) =P(q'),

whereP() is the topological pressure.

Conditions on'in the examples above are ment to ensure the absense of phase transition, i.e., existence and uniqueness of equilibrium state for potential q' for everyq2R. The main goal of this paper is to relax such conditions and to obtain

results for systems exhibiting phase transitions.

A natural class of observable functions ' would be the set of all continuous function. Moreoveor, the set of all continuous functions is quite rich in the sense of possible phase tranitions. For example 20, p.52], for any set f

1::: k g of

ergodic shift-invariant measures on AZ, where A is a nite set, one can nd a

continuous function'such that all these measuresi,i= 1::: k are equilibrium

states for '. Nevertheless, A.-H. Fan, D.-J. Feng in 7], and E. Olivier in 14], in the case of symbolic dynamics, obtained results on the spectrum E'() for

arbitrary continuous functions', similar to those mentioned above. In fact, they were studying the dimension spectrumD'(), but in symbolic case for everyone

hasE'() = #(A)D'(), where #(A) is the number of elements inA.

In this paper we study the entropy spectum E'() for a continuous

transformation f on a compact metric space (Xd) and arbitrary continuous function '. The main result of this paper (Theorem 5.1) states that if f is a continuous transformations with specication property, then for anywithK 6=?

one has

where H'() := sup n h(f) :is invariant and Z 'd=o 

and '() is a special \ball"-counting dimension ofK , similar to one introduced

in 7].

Readers, familiar with Large Deviations, will recognize in H'() the so-called

rate function. And indeed, we use the Large Deviation results for dynamical systems with specication obtained by L.-S.Young in 24].

The most intricate part of our proof is the equalityE'() = '(). To show it

we use a Moran fractal structure, inspired by one constructed in 7] for the symbolic case.

The Manneville-Pomeau map is a piecewise continuous map of a unit interval given by

fs(x) =x+x1+s mod 1 0< s <1:

This map has a unique indierent xed pointx= 0, and is probably the simplest example of a non-uniformly hyperbolic dynamical system. Thermodynamic properties of this transformation are quite well understood, see 19, 22, 12, 13].

In 18], M. Pollicott and H. Weiss studied the multifractal spectrum for' = logf0

s, i.e., the spectrum of Lyapunov exponents. They were able to obtain a partial

description of this spectrum. Using our results we able to complete the picture, see section 6 for details.

A straightforward modication of our proofs shows that the results are valid in more general settings as well. Supposef :X !X is a continuous transformation

with specication property and ' = ('1::: 'd) : X

! Rd is a continuous

function. For2Rd consider the set

K =n x2X: lim n!1 1 n n;1 X i=0 'j(fi(x)) =j j= 1::: d o : Then E'() =htop(fK ) = sup n h(f) :is invariant and Z 'd=o : (1)

In fact, even more is true. Suppose again that':X !Rd is a continuous function

and  : Im(')!Rmis a continuous map dened on Im(') =f'(x) :x2XgRd.

Dene K'  = n x2X : lim n!1 1 n n;1 X i=0 '(fi_(x)) =o :

Then for any such thatK'

 6=?one has E '() =htop(fK '  ) = sup n h(f) :is invariant and   Z 'd =o : (2)

As an imediate consequence of (1) and (2) we obtain the following result, which we call the Contraction Principle for Multifractal Spectra, due to the clear analogy with the well-knwon Contraction Principle in Large Deviations:

E

'() = sup :( )=

E'():

For more detailed discussion and some examples see section 7.

Everywhere in the present paper #(C) denotes a cardinality of a setC. Proofs of all lemmas are collected in section 8.

2 Multifractal spectrum of continuous functions

Letf :X !X be a continuous transformation of a compact metric space (Xd).

Throughout this paper we will assume thatf has nite topological entropy. Suppose

':X !Ris a continuous function. For2R dene:

K =n x2X: lim n!1 1 n n;1 X i=0 '(fi_{(x)) =}o : (3)

We introduce the following notation

L'=f2R: K 6=?g:

Lemma 2.1.

The setL' is a non-empty bounded subset of R.

Denition 2.1.

A continuous transformation f : X ! X satises specication

if for any " > 0 there exists an integer m = m(") such that for arbitrary nite intervals Ij = ajbj]N, j= 1::: k, such that

dist(IiIj)m(") i6=j

and anyx1::: xk in X there exists a point x

2X such that

d(fp+a jxfpx

j)< " for all p= 0::: bj;aj and every j = 1::: k:

Following the present day tradition we do not require that x is periodic. Specication implies topological mixing. Moreover, by the Blokh theorem 2], for continuous transformations of the interval these two conditions are equivalent. Using this equivalence and the results of Jakobson 9], we conclude that for the logistic family fr(x) = rx(1;x) the specication property holds for a set of

parameters of positive Lebesgue measure.

The specication property allows us to connect together arbitrary pieces of orbits. Suppose now that for two values1, 2 the corresponding sets K

1K 2 are not

empty. Using the specication property we are able to construct points with ergodic averages, converging to any number2(

12). Hence,

L' is a convex set. This

Lemma 2.2.

Iff :X !X satises specication, thenL' is an interval.

We recall that the entropy spectrum E'() of ' is the map assigning to each

2L'the value

E'() =htop(fK ): (4)

The denition and some fundamental facts about the topological entropy of non-compact sets are collected in the following section.

3 Topological entropy of non-compact sets

The generalization of the topological entropy to non-compact or non-invariant sets goes back to Bowen 3]. Later Pesin and Pitskel 17] generalized the notion of the topological pressure to the case of non-compact sets. In this paper we use an equivalent denition of the topological entropy, which can be found in 16].

3.1 Denition of the topological entropy.

Once again, let (Xd) be a compact metric space, andf :X !X be a continuous

transformation. For anyn2N we dene a new metricdn onX as follows:

dn(xy) = maxfd(fk(x)fk(y)) : k= 0::: n;1g

and for every" >0 we denote byBn(x") an open ball of radius"in the metricdn

aroundx, i.e.,

Bn(x") =fy2X : dn(xy)< "g:

Suppose we are given some set Z  X. Fix " > 0. We say that an at most

countable collection of balls ; = fBn i(xi")

gi coversZ if Z  iBn

i(xi"). For

; =fBn i(xi")

gi, putn(;) = minini. Lets0 and dene

m(ZsN") = inf

; X

i exp(

;sni)

where the innum is taken over all collections ; =fBn i(xi")

gcoveringZ and such

that n(;) N. The quantity m(ZsN") does not decrease withN, hence the

following limit exists

m(Zs") = lim_N

!1

m(ZsN") = sup_N>

0

m(ZsN"):

It is easy to show that there exists a critical value of the parameter s, which we will denote byhtop(fZ"), wherem(Zs") jumps from +1to 0, i.e.,

m(Zs") =

(

+1 s < htop(fZ")

There are no restriction on the value m(Zs") for s = htop(fZ"). It can be

innite, zero, or positive and nite. One can show 16] that the following limit exists

htop(fZ) = lim_"!0

htop(fZ"):

We will callhtop(fZ) the topological entropy of f restricted toZ, or, simply, the

topological entropy ofZ, when there is no confusion aboutf.

3.2 Properties of the topological entropy

Here we recall some of the basic properties and important results on the topological entropy of non-compact or non-invariant sets.

Theorem 3.1 (

16])

.

The topological entropy as dened above satises the following: 1. htop(fZ1) htop(fZ 2) for anyZ1 Z 2 X

2. htop(fZ) = sup_i htop(fZi), whereZ = 1

i=1Zi X

The next theorem establishes a relation between topological entropy of a set and the measure-theoretic entropies of measures, concentrated on this set, generalizing the classical result for compact sets.

Theorem 3.2 (

R. Bowen 3])

.

Letf :X !X be a continuous transformation of

a compact metric space. Suppose  is an invariant measure, and Z X is such

that(Z) = 1, then

htop(fZ)h(f)

whereh(f) is the measure-theoretic entropy.

Suppose we are given an invariant measure. A pointx is called generic for

if the sequence of probability measures

xn= 1_nn ;1 X k=0 fk (x)

whereyis the Dirac measure aty, converges toin the weak topology. Denote by G the set of all generic points for. Ifis an ergodic invariant measure, then by

the Ergodic Theorem(G) = 1. Applying the previous theorem we immediately

conclude thathtop(fG)h(f). In fact, the opposite inequality is valid as well:

Theorem 3.3 (

R. Bowen 3])

.

Letbe an ergodic invariant measure, then

htop(fG) =h(f):

Ya. Pesin and B. Pitskel in 17] have proved the following variational principle for non-compact sets.

Theorem 3.4.

Suppose f :X !X is a continuous transformation of a compact

metric space(Xd), and Z X is an invariant set. Denote byMf(Z) the set of

all invariant measures such that (Z) = 1. For any x2X denote by V(x) the

set of all limit points of the sequencefxng. Assume that for every x2Z one has

V(x)\Mf(Z)6=?:

Then htop(fZ) = sup_

2Mf(Z)

h(f).

The conditions of this theorem are very dicult to check in any specic situation. However, there is no hope for improving the above result for general setsZ. There are examples 16, 17] of sets, for which the conditionV(x)\Mf(Z)6=?does not

hold for allx2Z, and one has a strict inequality

htop(fZ)>supfh(f) : 2Mf(X) and(Z) = 1g:

In this paper we restrict our attention to the sets of a special form: namely, the setsK given by (3). For these particular sets we prove a variational priciple for the topological entropy, provided the transformationf satises specication:

Theorem 3.5.

Suppose f : X ! X is a continuous transformation with the

specication porperty. Let'2C(XR) and assume that for some 2R

K =n x2X: lim n!1 1 n n;1 X i=0 '(fi_{(x)) =}o 6 =? then htop(fK ) = sup n h(f) :is invariant and Z 'd=o :

Remark 3.1.

Under the conditions of the above theorem, it is possible that for a certain parameter value , there exists a unique invariant probability measure 

withR

'd=, such that

htop(fK ) =h(f):

Hence,  is a measure of maximal entropy among all invariant measures  with

R

'd = . However, it is also possible, that  (K ) = 0. This situation, for example, occurs in the family of Manneville-Pomeau maps, see Remark

??

for more details.

3.3 Entropy distribution principle.

The following statement will allow us to estimate the topological entropies of the sets from bellow, without constructing probability measures, which are invariant and concentrated on a given set. It is sucient to consider only probability measures, which need not be invariant, but which satisfy some specic 'uniformity condition'. We call this result the Entropy Distribution Principle, by the clear analogy with a well-known Mass Distribution Principle 6].

Theorem 3.6 (Entropy distribution principle).

Let f : X ! X be a

continuous transformation. Suppose a set Z X and a constant s 0 are such

that for any" >0 one can nd a Borel probability measure=" satisfying

1) "(Z)>0,

2) "(Bn(x"))C(")e

;ns for some constantC(")>0 and every ball

Bn(x")

such that Bn(x")\Z 6=?.

Then htop(fZ)s.

Proof. We are going to show thathtop(fZ")sfor every suciently small" >0.

Indeed, choose such" >0 and consider the corresponding probability measure".

Let ; = fBn i(xi")

gi be some cover of Z. Without loss of generality we may

assume thatBn i(xi")

\Z 6=?for everyi. Then X i exp( ;sni)C(") ;1 X i "( Bn i(xi")) C(") ;1 "  iBn i(xi")  C(") ;1 "(Z)>0:

Thereforem(Zs")>0, and hencehtop(fZ")s.

4 Upper estimates of

E '

( )

.

In this section we are going to dene two auxiliary quantitiesH'() and '().

These quantities will be used to give an upper estimate on the multifractal spectrum

E'().

4.1 Denition of

H '

( )

Let us introduce some notation

M(X) : the set of all Borel probability measures onX

Mf(X) : the set of allf-invariant Borel probability measures onX

Mef(X) : the set of all ergodicf-invariant Borel probability measures onX Mf(X') : the set of allf-invariant Borel probability measures, such that

Z

'd=:

We consider the weak topology onM(X) and also on its subsetsMf(X),Mef(X),

etc.! as it is well known,M(X) is compact metrizable space in the weak topology.

Lemma 4.1.

For any  2 L' the set Mf(X') is a non-empty, convex and

closed (in the weak topology) subset of Mf(X).

This result allows us to dene the following quantity: for any2L' put

H'() = sup

n

h(f) : 2Mf(X') o

Lemma 4.2.

For any '2 C(XR), H'() is a concave function on the convex

hull of L'.

4.2 Denition of

 '

( )

Here, following the approach of 7], we introduce another dimension-like characteristic '() of the setK . We use a word \dimension" in association with

'(), because '() is dened in terms similar to the denition of Hausdor or

box counting dimensions.

For2L'and any >0 andn2N put

P(n) =n x2X :    1 n n;1 X i=0 '(fi_(x));   <  o :

Clearly, for2 L' and any > 0 the set P(n) is not empty for suciently

largen.

Fix some " > 0 and let N(n") be the minimal number of balls Bn(x"),

which is necessary for covering the set P(n). (If P(n) is empty we let

N(n") = 1).

Obviously, N(n") does not increase if  decreases, and N(n") does not decrease if " decreases. This observation guarantees that the following limit exists '() = lim_" !0 lim !0 lim n!1 1 nlogN(n"): (6)

One can give another equivalent denition of '(). The equivalence of these

denitions will be useful for subsequent arguments. Let us recall a notion of (n" )-separated sets: a set E is called (n")-separated if for any xy 2 E, x 6= y,

dn(xy)> ".

By denition, we letM(n") be the cardinality of a maximal (n")-separated set inP(n). Again, we putM(n") = 1 ifP(n) is empty. A standard argument shows that

N(n")M(n")N(n"=2) (7)

for everyn2N and all", >0.

Moreover, iff satises specication, then taking an upper limit instead of the lower limit with respect tonin the denition of '() will give the same number.

Lemma 4.3.

Iff satises specication, then

'() = lim_" !0 lim !0 lim n!1 1 nlogN(n"):= lim"!0 lim !0 lim n!1 1 nlogM(n"):

We will not use this result, and therefore, will not give a proof, which is based on establishing some sort of subadditivity ofN(n"):

(N(n4"))kN(4nk+km(")")

for all integersk1 and all suciently largen, wheremis taken from the denition

4.3 Upper estimate for

E ' ( )

in terms of

H ' ( )

via

 ' ( )

.

Theorem 4.1.

For any2L' one has

E'()'()H'():

Proof. The rst inequality E'()  '() is quite easy. Its proof is based on a

standard \box-counting" argument. Following 7], for 2 L',  >0 and k 2 N

consider sets G(k) = 1 \ n=k P(n) = 1 \ n=k n x2X:    1 n n;1 X i=0 '(fi_(x));   <  o :

It is clear, that for any >0

K =n x2X : lim n!1 1 n n;1 X i=0 '(fi_{(x)) =}o  1  k=1 G(k): (8) We are going to show that htop(fG(k)")  '() holds for any k  1,

implyinghtop(fK ")'() as well.

Fix arbitraryk1, thenG(k) (as a subset of P(n) for nk) can be

covered byN(n") ballsBn(x") for allnk. Therefore for everys0 and

allnk we have

m(G(k)s")N(n")exp(;ns): (9)

Suppose now thats >'(), and put = (s;'())=2>0. Since

'() = lim_" !0 lim !0 lim n!1 1 nlogN(n")

for all suciently small " > 0 and  > 0, there exists a monotonic sequence of integersnl!1such that

N(nl")exp ;

nl('() + )

for all l1.Without loss of generality we may assume that n 1

 k. Then, from

(9) we obtain

m(G(k)s")exp(;nl )

and hencem(G(k)s") = 0. Thereforehtop(fG(k)")s, and

htop(fK ")sup

k htop(fG(k)")s

due to (8). Thererefore,htop(fK ) = lim"!0htop(fK ")

 sas well. Finally,

sinces > '() was chosen arbitrary, we conclude that E'() :=htop(fK ) 

'().

The second inequality '()H'() is closely related to the second statement

last stage of our proof, similar to 24], we will rely on one fact, which is established in a standard proof of the variational principle for the classical topological entropy 23].

In order to show the inequaity '() H'(), it is sucient, for any >0,

to present a measure2Mf(X') (i.e., an invariant measure with R

'd=) such that

h(f)'(); :

Fix arbitrary > 0. By the denition of '(), there exists a suciently small "0>0 such that for all"

2(0" 0) one has '(") = lim_ !0 lim n!1 1 nlogN(n")>'(); 1 3 : Put "k = "0

2k, k 1. For any k1 one can nd a suciently small k, k !0,

such that lim n!1 1 nlogN(kn"k)>'(); 2 3 : Also, for anyk1 we choose somenk 2N, nk!1, such that

Nk :=N(knk"k)>exp;

nk('(); )

:

LetCk be the centers of some minimal covering ofP(knk) by ballsBn

k(x"k).

Note, that #(Ck) = Nk, and Bn k(x"k)

\P(knk) 6= ? for every x 2 Ck.

Otherwise, the covering, would not be minimal. For anyk1 dene a probability

measure k = 1_Nk X x2C k x and let k = 1_nk n k ;1 X i=0 (f;i) k = 1_Nk X x2C k 1 nk nk ;1 X i=0 fi (x):

Let  be some limit point for the sequence k. By Theorem 6.9 in 23],  is an

invariant measure, and we claim that

Z

'd= (10)

i.e.,2Mf(X'). Indeed, for everyk1, one has    Z 'dk;     1 Nk X x2C k    1 nk nk;1 X i=0 '(fi_(x));   :

However, for every x 2 Ck there exists y = y(x) 2 P(knk) such that

dnk(xy)< "k. Therefore    1 nk nk ;1 X i=0 '(fi_(x));    1 nk nk ;1 X i=0  '(fi(x)) ;'(fi(y))  +k Var('"k) +k

where Var('"k) = sup;

j'(x);'(y)j : d(xy) < "k

! 0 as k !1, since ' is

continuous. Hence, we conclude that

Z

'dk ! k!1:

The above invariant measureis a limit point for the sequencek. Hence, there

exists a sequencekj!1such thatk j

!weakly. This in particular means that Z

'dkj !

Z

'd:

Therefore we obtain (10). Finally, repeating the second half of the proof of the classical variational principle 23, Theorem 8.6, p. 189-190] we conclude that

h(f) lim k!1 1 nk logNk  lim k!1 1 nk logNk'(); :

This nishes the proof.

5 Lower estimate on

E '

( )

.

The main result of this section is the following theorem.

Theorem 5.1.

Let f : X ! X be a continuous transformation with the

specication property and'2C(XR). Then for any 2L' one has

E'() = '() =H'(): (11)

Proof. In Theorem 4.1 we proved that for any continuous transformationf one has

E'()  '()  H'() for all  2 L'. Hence, it is sucient for the proof of

(11) to show the opposite inequalitiesE'()'()H'(). We start with the

inequality '()H'(). Our proof relies on the proof of statement 3 of Theorem

1 in 24], but let us rst recall one result of A. Katok 11].

Theorem 5.2.

Let f : X ! X be a continuous transformation on a compact

metric space, and  be an ergodic invariant measure. For " > 0,  > 0 denote byN _f("n) the minimal number of "-balls in thedn-metric which cover a set of

measure at least 1;. Then, for each 2(01), we have

h (f) = lim_"!0 lim n!1 1 nlogN f("n) = lim"!0 lim n!1 1 nlogN f("n):

Remark 5.1.

Suppose  is ergodic andY X is such, that(Y)1;. Denote

byS(Y"n) the maximal cardinality of an (n")-separated set inY. Similar to(7) we conclude thatS(Y"n)N f("n).

To prove the inequality '()  H'(), it is sucient to show that for any

>0 and every2Mf(X') one has

'()h(f);4 :

1) > ! 2) d(xy)< " ) j'(x);'(y)j< ! 3) lim_n !1 1 nlogN(3n")<'() + .

We can approximateby an invariant measurewith the following properties (see 24, p.535]):

a)  =Xk

i=1

ii, wherei>0,P

ii= 1, andi is an ergodic invariant measure

for everyi= 1::: k! b) h (f)h(f); ! c)    Z 'd; Z 'd  < .

Sincei is ergodic for every i, there exists a suciently large N such that the

set of points Yi(N) = n x2X :    1 n n;1 X j=0 '(fj_(x)); Z 'di   < for alln > N o

has ai-measure at least 1; for everyi= 1::: k.

Therefore, according to Theorem 5.2, there exist integersNi such that for all ni> Ni the minimal number of 4"-balls indni-metric, which is necessary to cover

Yi(N) is greater than or equal to exp(ni(h i(f)

; )). This implies, according to

the remark 5.1, that the cardinality of a maximal (ni4")-separated set in Yi(N)

is greater than or equal to exp(ni(h i(f)

; )). Finally, choose a suceintly large

integerN0 such that for everyn > N0 one has

ni:= in]>max(NiN)

for all i = 1::: k, also denote by C(ni4") some maximal (ni4")-separated

set in Yi(N). For every k-tuple (x1::: xk), where xi

2 C(ni4"), nd a point

y =y(x1::: xk)

2X such that it shadows pieces of orbits fxi::: fn i;1x

iji =

1::: kgwithin the distance"and the gapm=m("). Put ^n=m(k;1) + P

ini.

Firstly, we observe that to dierent (x1::: xk) 2 Cn

1

:::Cn

k correspond

dierent points y = y(x1::: xk). This is indeed the case, because for y =

y(x1::: xk) andy 0=y(x0 1::: x 0 k) one has dn^(yy 0)>2": (12)

Secondly, for everyy =y(x1::: xk) one has    1 ^ n ^ n;1 X p=0 '(fp_(y));   <2+ km ^ n jj'jjC 0:

Hence, for suciently large ^n (i.e., large n) every point y = y(x1::: xk) is in

P(3n^).

On the other hand, due to (12), one would need at least #(Cn1) :::#(Cn k) exp  1n](h 1(f) ; ) +:::+ kn](h k(f) ; )  exp ; n(h (f);2 ) exp ; n(h(f);3 )

"-balls in thedn^-metric to coverP(3n^). Therefore

lim

n!1

1 ^

nlogN(3n"^ )h(f);3 :

Hence, due to the choice of" >0, we have '()+ > h(f);3 . This nishes

the proof of our rst inequality '()H'().

A much more dicult inequality to prove is the the remaining one: E'() 

'(). In order to show it we will construct a Moran fractal, suitable for the

purposes of computation of topological entropy. Roughly speaking Moran fractal is a limit set of a following geometric construction: consider a monotonic sequence of compact setsfFkg, Fk

+1

Fk, such thatFk is a union ofNk closed sets # (k)

i , i= 1::: Nk, of approximately the same size. Moreover, the sets #(k+1)

i forming

the (k+ 1)-level of the construction are somewhat similar to the sets #(k)

i of the k-th level. The Moran fractal associated to this consturction is the setF

F =\

k Fk:

One could think of a Moran fractal as a generalization of a standard middle-third Cantor set. A particular choice ofFkwill ensure that the limit setF will be a closed

subset of K , but also will allow us to construct a probability measure  on F, satisfying the conditions of the Entropy Distribution Principle withs= '();

for any > 0. Thus the topological entropy of F will be larger or equal than s. SinceFK , the same will be true for the topological entropy of K .

Fix some >0, and choose a suciently small" >0 such that lim !0 lim n!1 1 nlogM(n8")'(); =2:

We assumed that f satises specication, let m=m(") be as in the denition of the specication property, and let

mk=m("=2k) k1:

Choose also some sequencek#0 and a sequencenk"+1such that

Mk:=M(knk8")>exp;

nk('(); )

 and nk 2m k:

By denition Mk is the cardinality of a maximal (nk8")-separated set in P(knk). Denote byCk =fxkiji= 1::: Mkgone of these maximal (nk8"

)-separated sets.

Step 1. Construction of intermidiate sets

Dk

.

We start by choosing

some sequence of integersfNkgsuch thatN

1= 1 and two following conditions are

satised: 1) Nk 2n k +1 +m k +1 fork 2! 2) Nk+1 2N 1n1 +:::+N k (n k +m k ) fork 1.

Then this sequenceNk is growing very fast, and in particular

lim k!1 nk+1+mk+1 Nk = 0 and limk!1 N1n1+:::+Nk(nk+mk) Nk+1 = 0: (13) For anyNk-tuple (i1::: iN

k)

2f1::: MkgN

klety(i

1::: iN

k) be some point

which shadows pieces of orbitsfxki

jfxkij::: f nk ;1xki j g,j= 1::: Nk, with a gap mk, i.e., dnk(xijf ajy(i 1::: iN k))< "2k

where aj = (nk +mk)(j ;1), j = 1::: Nk. Such point y(i

1::: iN

k) exists,

becausef satises specication. Collect all such points into the set

Dk= y(i1::: iN k) j i 1::: iN k 2f1::: Mkg  : (14)

We claim that dierent tuples (i1::: iN

k) produce dierent points

y(i1::: iN

k), and that these points are sucienly separated in the metric dtk,

where

tk=Nknk+ (Nk;1)mk:

This is the content of the following lemma.

Lemma 5.1.

If(i1::: iN k) 6 = (j1::: jN k), then dtk(y(i 1::: iN k)y(j 1::: jN k))>6": (15) Hence,#(Dk) =MNk k .

SinceN1= 1, without loss of generality we may assume thatD1=C1.

Step 2. Construction of

Lk

.

Here we construct inductively a sequence of

nite setsLk. Points ofLk will be the centers of a balls forming thek-th level of

our Moran construction.

LetL1 =D1 and putl1=n1. Suppose we have already dened a setLk, now

we present a construction ofLk+1. We let

lk+1=lk+mk+1+tk+1=N1n1+N2(n2+m2) +:::+Nk+1(nk+1+mk+1):

For everyx2Lk andy2Dk

+1 letz=z(xy) be some point such that

dlk(xz)< "2k +1 and dt k +1(yf lk+m k +1z)< " 2k+1: (17)

Such a point exists due to the specication property off. Collect all these points into the set

Lk+1= n z=z(xy)jx2Lk y2Dk +1 o : (18)

Similar to the proof of Lemma 5.1 we can show that dierent pairs (xy),x2Lk,

y2Dk

+1, produce dierent pointsz=z(xy). Hence, #(Lk+1) = #(Lk)#(Dk+1).

Therefore, by induction #(Lk) = #(D1):::#(Dk) =M N1 1 :::M Nk k :

It immediately follows from (15) and (17), that for every x 2 Lk and any

yy0 2Dk +1,y 6 =y0, one has dlk(z(xy)z(xy 0))< "

2k and dlk +1(z(xy)z(xy

0))>5": (19)

There is an obvious tree structure in the construction of the setsLk. We will

say that a pointz2Lk

+1 descends fromx

2Lk if there existsy2Dk

+1 such that

z=z(xy). We also say that a pointz2Lk

+p descendsfromx

2Lkif there exists

a sequence of points (zk::: zk+p),zk =x, zk+p=z, and zt

2Lt, such thatzl +1

descends fromzlin the above sense for everyl=k::: k+p;1.

Step 3. The Moran fractal

F

.

For everykput

Fk =  x2L k Bl k  x "_2k;1  

whereBl(x) is the closed ball aroundxof radiusin the metricdl, i.e., Bl(x) =fy2X : dl(xy)g:

Lemma 5.2.

For everyk the following is satised: 1) for anyxx0 2Lk,x6=x 0, the sets Bl k  x "_2k;1  ,Bl k  x0 " 2k;1  are disjoint 2) if z2Lk +1 descends from x 2Lk, then Bl k +1  z "_2k Bl k  x "_2k;1  : Hence,Fk+1 Fk. Finally, we put F = \ k1 Fk:

Lemma 5.3.

For everyx2F one has lim n!1 1 n n;1 X k=0 '(fi_{(x)) =:} ThereforeF K .

Step 4. A special probability measure



.

For everyk1 dene an atomic

probability measurek as follows k(fzg) = 1

#(Lk) for everyz2Lk:

Obviously,k(Fk) = 1.

Lemma 5.4.

A sequence of probability measuresfkgconverges in a weak topology.

Denote the limiting measure by , then(F) = 1.

An important property of the limiting measureis formulated in the next lemma.

Lemma 5.5.

For every suciently largenand every pointx2X such that Bn(x"=2)\F6=?

one has

(Bn(x"=2))e

;n(s;): (20)

Summarizing all from above we see that for every positive and every suciently small" >0, we have constructed a compact setF,F K , and a measuresuch

that (20) holds. >From the Entropy Distribution Principle and the fact thatK ,

we conclude '();2 =s; htop(fF"=2)htop(fK "=2) and hence E'() =htop(fK ) = lim "!0 htop(fK ")'();2 :

Since > 0 is arbitrary, we nally conclude that E'()  '(), which nishes

the proof of Theorem 5.1.

6 Manneville-Pomeau map

Before we start we the detailed discussion of the multifractal spectrum for Lyapunov exponents of the Manneville-Pomeau maps, let us establish a general relation between the multifractal spectra in general and the Legendre transform of the pressure function.

For a continuous function':X !R, andq2R letP'(q) =P(q'), whereP()

is the topological pressure. By the classical Variational Principle one has

P() = supn h(f) + Z d: 2Mf(X) o :

Since we have assumed that the topological entropy off is nite,P() is nite for every continuous. Moreover,P() is convex, Lipschitz continuous, increasing and

P(c++;f) =c+P(), wheneverc2R, and 2C(XR).

For any2Rdene the Legendre transformP  '() by P '() = inf_q2R  P'(q);q  : Note, thatP

'()<+1for all2R, however, it is possible thatP 

'() =;1.

Theorem 6.1.

Letf :X !X be a continuous transformation with specication,

and':X !R be a continuous function. Then

(i) for any 2L', one has

H'()P 

'()!

(ii) if, moreover, f is such that the entropy map  ! h(f) is upper

semi-continuous, then for any from the interior ofL' one has

H'() =P

'():

Remark 6.1.

Transformationsf :X !X with an upper semi-continuous entropy

map

H() :Mf(X)!0+1) : !h(f)

play a special role in the theory of equilibrium states. This class of transformations includes, for example, all expansive maps 23]. A useful property of such transfomations is that every continuous functionhas a least one equilibrium state. Proof of Theorem 6.1. (i) For any2L' and anyq2Rone has

H'() = sup n h(f) : 2Mf(X) Z 'd=o = supn h(f) +q Z 'd: 2Mf(X) Z 'd=o ;q sup n h(f) +q Z 'd: 2Mf(X) o ;q=P(q');q

where the last equality follows to the Variational Principle for topological pressure. Hence,H'()infq ; P(q');q  =P '().

(ii) It was shown by O. Jenkinson 10], that if the entropy map is upper semi-continuous, then for any  from the interior of L', there exists q



2 R and an

invariant measure, which is an equilibrium state forq'such that Z

Hence H'() = sup n h(f) : 2Mf(X) Z 'd=o h (f) =P(q ') ;q : ThereforeH'()P 

'() and the result follows.

The following theorem is an immediate corollary of Theorems 5.1 and 6.1.

Theorem 6.2.

Suppose f : X ! X is a continuous transformation with

specication property such that the entropy map is upper semi-continuous. Then for any2(infL'supL') one has

E'() =P 

'():

Remark 6.2.

Note that for transformations with the specication property, L' is

an interval.

Let us consider in greater detail an application of the above theorem to the multifractal analysis of the Manneville-Pomeau maps.

For a given numbers, 0< s < 1, a corresponding Manneville-Pomeau map is given by

f : 01]!01] :x!x+x

1+s mod 1:

The map f is topologically conjugated to a one-sided shift on two symbols, and thus satises the specication property. Morevoer,f is expansive, and hence the entropy map is upper semi-continuous. Let'(x) = logf0(x). With such choice the

level setsK are preciesly the level sets of pointwise Lyapunov exponents, which are dened (provided the limit exists, of course) as

(x) = lim_n !1 1 nlogj(fn) 0(x) j and K = x: (x) = :

Due to the fact thatx= 0 is an indierent xed point for the Manneville-Pomeau map, there exist pointsxwith(x) arbitrary close to 0, and hence infL'= 0.

Let us discuss some thermodynamic properties of the Manneville-Pomeau maps. First of all, there exists a unique absolutely continuous f-invariant measure . Moreover,is an equilibrium state for the potential;'andis ergodic. However,

there exists another equilibrium state for;', namely, the Dirac masure at 0,  0.

The coexistence of two equilibrium states results in a non-analytic behaviour of the pressure functionP'(q) :=P(q'). Namely, it was shown in 19, 22] that P'(q) is

positive and strictly convex forq >;1, andP'(q)0 forq;1, see Figure 1.

Since f satises specication and is expansive, Theorem 6.2 is applicable and henceE'() =P



'(). The graph ofP

'() is shown in Figure 1.

The entropy spectrumE'() is concave, but not strictly concave. The graph of E'() contains a piece of a straight line.

We represent the interval infL'supL'] = 0%] as the union of two intervals

00] and (0%], where 0 is the largest  such that P

() = , i.e., P( ) is linear on 00]. In fact, 0=h(f) = Z logf0d

P(q)

q=-1 q

α)

α0 α

P*(

Figure 1: The pressure functionP'(q) and its Legendre transformP() = E'().

whereis an absolutely continuous invariant measure. Additional considerations show that:

 For each  2 (0

0) there exists a unique invariant measure  2 Mf(01]') such that h(f) = sup n h (f) : is invariant and Z 'd=o 

i.e., is a measure of maximal entropy inMf(01]'), and hence

htop(fK ) =h(f)!  Moreover, for any2(0

0) one has

 =+ (1;) 0

whereis the absolutely continuous invariant measure mentioned above. Since,0 are ergodic, andK are invariant sets, we conclude that

 (K ) = 0 for all2(0

0). This is a new phenomenon, because until a typical situation in

multifractal analysis would be  (K ) = 1 for the \maximal" measure  . And indeed, for all2(

01], the measures of maximal entropy in

Mf(01]')

exist as well, but

 (K ) = 1:

The explanation of this phenomenon lies in fact that the pressure function has a phase transition of the rst order atq=;1.

7 Multidimensional spectra and Contraction Principle

Suppose f : X ! X is a continuous transformation of a compact metric space

(Xd) satisfying specication property, and':X !Rd is a continuous function.

Suppose also that we are given a continuous map  :U !Rm:

whereU Rd is such that Im(') =

'(x) : x2X 

U. For any 2Rm dene

a set K'() = n x2X : lim n!1 1 n(Sn')  =o :

We are interested in the entropy spectrum of ', i.e., the function E '() =htop ; fK'()  dened on a setL '= f : K '() 6 =?g:Our claim is

Theorem 7.1.

Let f be a continuous transformation satisfying the specication property, and':X!Rd :Rd !Rm be continuous map such that&'is well

dened. Then that for every2L

' one has E '() = sup n h(f) :  is invariant and   Z 'd =o : (21)

The proof of this fact is a generalization of the 1-dimensional proof presented in the previous sections.

We would like to discuss now some corollaries of Theorem 7.1. First of all, by taking  to be identity we immediately conclude that

E'() =htop(fK') = sup n h(f) :  is invariant and Z 'd=o : (22) A second corollary is the following theorem, which we call the

Contraction

Principle for entropy spectra

due to a clear analogy to a well-known Contraction Principle from the theory of Large Deviations, see e.q. 5].

Theorem 7.2.

Under conditions of Theorem 7.1, for any2L

' one has E

'() = sup :( )=

E'(): (23)

Proof. The statement follows from the variational descriptions (21), (22) of the entropy spectra E

'() and

E'(). Indeed, to prove the claim we have to show

that supn h(f) : 2Mf(X) and   Z 'd =o = sup :( )= supn h(f) : 2Mf(X) and Z 'd=o : (24)

A proof of (24) is straightforward.

In our opinion, it is an interesting question whether the contraction principle (23) is valid for systems without specication.

For transformationsf with the specication property the domainL'is a convex

set, and E'() is a concave function. Theorems 7.1, 7.2 can be used to produce

multifractal spectra E

' which are not concave, or dened on a non-convex

domainsL

'. For another setup which also leads to a non-concave multifractal

spectra see 1, Proposition 10].

8 Proofs

Proof of Lemma 2.1. Any continuous transformation of a compact metric space admits an invariant probability measure. Moreover, there exist ergodic invariant measures. Supposeis ergodic, then by Ergodic Theorem

1 n n;1 X i=0 '(fi_(x))! Z 'd as n!1

for -a.e. x 2 X. Hence, L' 6= ?. Clearly, L'   ;jj'jjC0jj'jjC0 , where jj'jjC 0 = maxx j'(x)j<1.

Proof of Lemma 2.2. Suppose K i

6

= ?, i = 12. let t 2 (01) and put  =

t1+ (1 ;t)

2. Choose somexi 2K

i and take any i

2V(xi), i= 12, where

V(x) is the set of limit points for the sequence of probability measure

xn= 1_nn ;1 X k=0 fk (x):

Theniis an invariant measure withR

'di=i,i= 12 (see the proof of Lemma

4.1 below). Put  = t1+ (1 ;)

2. Obviously, R

'd = . Now, we apply 4, Proposition 21.14], which says that for a transformation with the specication property every invariant measure (not, necessarily ergodic!) has a generic point, i.e., there exists a pointx2X such thatxn!asn!1. Hence, for the same

pointxone has

Z 'dxn= 1_nn ;1 X i=0 '(fi(x))! Z 'd= and therefore,K 6=?.

Proof of Lemma 4.1. We start by showing that Mf(X') is not empty for any

2L'. Take any x 2K , and denote byV(x) the set of all limit points of the

sequence fxngn

1. Due to compactness of

M(X) the set V(x) is not empty.

 2 V(x). By the construction of V(x), there exists a sequence nk ! 1 such thatxnk !weakly. Hence 1 nk nk ;1 X i=0 '(fi_(x))! Z 'd k!1:

Sincex2K , we obtain that R

'd=, and hence, 2Mf(X'). Convexity

and closedness ofMf(X') are trivial.

Proof of Lemma 4.2. Convexity ofH'() is an obvious consequence of the anity

of the entropy maph(f) :Mf(X)!0+1], 4].

Proof of Lemma 5.1. If (i1::: iN k)

6

= (j1::: jN

k), there existlsuch thatil

6 =jl. By the construction ofy(i1::: iN k) andy(j 1::: jN k) we have dnk(xki lf aly(i 1::: iN k))< " and dnk(xkj lf aly(j 1::: jN k))< ":

Sincexkilxkjl are dierent points in the (nk8")-separated set, one has

dnk(f aly(i 1::: iN k)f aly(j 1::: jN k)) dn k(xki lxkjl) ;dn k(xki lf aly(i 1::: iN k)) ;dn k(xkj lf aly(j 1::: jN k)) >8";";"= 6": Since dtk(y(i 1::: iN k)y(j 1::: jN k)) dn k(f aly(i 1::: iN k)f aly(j 1::: jN k))

the proof is nished.

Proof of Lemma 5.2. 1) By (19) for xx0

2 Lk, x 6= x 0, one has d lk(xx 0) > 5". Hence Bl k  x "_2k;1  \ Bl k  x0 " 2k;1  =?: 2) For x 2 Lk and z 2 Lk

+1 such that z descends from x, by (19) one has

dlk(xz) < "=2 k_{. Hence,} Bl k(z"=2 k₎  Bl k(x"=2 k;1). Finally, sincel k+1 > lk, one has Bl k +1(z"=2 k₎Bl k(z"=2 k_): Proof of Lemma 5.3.

Estimate on

Dk

.

Let us introduce some notation: for anyc >0 put

Var('c) = supfj'(x);'(y)j: d(xy)< cg:

Note, that due to compactness of X, Var('c)! 0 as c !0 for any continuous

function'. Also, ifdn(xy)< c, then

   n;1 X i=0 '(fi_(x)); n;1 X i=0 '(fi_(y))   n;1 X i=0   '(fi(x));'(fi(y))   nVar('c):

Suppose now that y 2 Dk, let us estimate   Pt k;1 p=0 '(f p_(y));tk  . By the

denition of Dk, there exist a Nk-tuple (i1::: iN

k), and points xki j 2 Ck for j= 1:::Nk, such that dnk(xki jf ajy)< " 2k whereaj= (nk+mk)(j;1). Hence,    nk ;1 X p=0 '(fp_xki j) ; nk ;1 X p=0 '(faj +py)    nkVar ; ' "_2k : Sincexkij 2CkP(knk) we have    nk ;1 X p=0 '(faj +py) ;nk    nk  Var; ' "_2k +k  : (25) To estimate  Pt k ;1 p=0 '(f p_(y));tk 

we represent the interval 0tk

;1] as the union Nk;1  j=0 ajaj+nk;1] N k;2  j=0 aj+nkaj+nk+mk;1]:

On the intervals ajaj+nk;1] we will use the estimate (25), and on the intervals

aj+nkaj+nk+mk;1] we use that    mk ;1 X p=0 '(faj+n k +py) ;mk    mk(jj'jjC0+jj)2mkjj'jjC0 since2L'  ;jj'jjC0jj'jjC0 . Therefore    tk ;1 X p=0 '(fp_(y));tk    Nknk  Var(' "_2k) +k  + 2(Nk;1)mkjj'jjC0: (26)

Estimate on

Lk

.

Introduce Rk = max_z 2L k    lk ;1 X p=0 '(fp_(z));lk   :

Let us obtain by induction an upper estimate onRk.

Ifk = 1, thenL1 =D1 =C1

P(

1n1) (note, that l1 =n1), therefore we

have

R1 l

11:

By the dention ofLk+1 everyz 2Lk

+1 is obtained by shadowing of some points

x2Lk andy2Dk +1: dlk(xz)< "2k +1 dt k +1(yf lk +m k +1z)< " 2k+1:

Hence,    lk +1 ;1 X p=0 '(fp_(z));lk +1       lk;1 X p=0 '(fp_(z)); lk;1 X p=0 '(fp_(x))  +    lk;1 X p=0 '(fp_(x));lk    +   lk +m k +1 X p=l k '(fp_(z));mk +1    +   tk +1 ;1 X p=0 '(flk +m k +1 +p(z)) ; tk +1 ;1 X p=0 '(fp(y))  +    tk +1 ;1 X p=0 '(fp(y));tk +1    lkVar ; ' "_2k+1 +Rk+ 2mk+1 jj'jjC0+tk +1Var ; ' "_2k+1 +Nk+1nk+1  Var(' "_2k+1) +k +1  + 2(Nk+1 ;1)mk +1 jj'jjC0

where we have used the estimate (26) for  Pt k +1;1 p=0 '(f p_(y));tk +1  . Hence Rk+1 Rk+ 2lk +1Var(' " 2k+1) +lk +1k+1+ 2Nk+1mk+1 jj'jjC 0 and by induction Rk2 k X p=1 lp  Var; ' "_2p +p+Np_lpmpjj'jjC0  : (27)

Let us analyse the obtained expression forRk. We claim that Rk=lk ! 0 as

k ! 1. We start by observing that, Var ; ' " 2 k ! 0 since ' is continuous.

By the choice of the sequence fkg one has k ! 0 as well. Moreover, since

lk  Nk(nk+mk) and the sequencefnkg is such that nk ! 1 ask ! 1, and

nk2m

k, we conclude thatmk=nk

!0 as well. Therefore, we can rewrite (27) as

Rk 

k

X

p=1

lpcp

whereck !0 ask!1. By the choice ofNk (13), we havelk 2l

k ;1, hence for

suently largekone has

Rk lk ck+ 1 k k;1 X p=1 cp and henceRk=lk!0 ask!1.

Estimate on

F

.

Now, supposex2F,n2N andn > l

1. Then there exists a

uniquek1 such that

lk < nlk +1:

Also, there exist a uniquej, 0jNk +1

;1 such that

lk+j(nk+1+mk+1)< n

lk+ (j+ 1)(nk

Sincex2F there existsz2Lk

+1 such that

dlk +1(xz)< "2k:

On the other hand sincez2Lk

+1 there exist %x 2Lk andy2Dk +1 such that dlk(%xz)< "2k +1 dt k +1(yf lk +m k +1z)< " 2k+1: Therefore dlk(xx%)< "2k ;1 dt k +1(f lk +m k +1xy)< " 2k;1:

Moreover, if j > 0, then by the denition of Dk+1 there exist points

xk+1 i1 ::: x k+1 ij 2Ck +1 such that dnk +1(x k+1 it f aty)< " 2k+1 whereat= (nk+1+mk+1)(t ;1), t= 1::: j, and hence dnk +1(x k+1 it f lk +m k +1 +a tx)< " 2k;2: (28)

We represent 0n;1] as the union

0lk;1] j t=1 lk+ (t;1)(mk +1+nk+1)lk+t(mk+1+nk+1) ;1]  lk+j(mk+1+nk+1)n ;1]: One has    lk ;1 X p=0 '(fp_x);lk        lk ;1 X p=0 '(fp_x); lk ;1 X p=0 '(fp_x%)  +    lk ;1 X p=0 '(fp_x%);lk    lkVar ; ' "_2k;1 +Rk

On each of the intervals atat+(mk+1+nk+1)

;1], whereat=lk+(t;1)(mk +1+ nk+1), we estimate    at +m k +1 +n k +1 ;1 X p=a t '(fp_x);(mk +1+nk+1)    2mk +1 jj'jjC 0+nk +1k+1+nk+1Var('"=2 k;2)

because of (28) and the fact thatxk+1

ij 2Ck +1 P(k +1nk+1). Finally, on lk+j(mk+1+nk+1)n ;1] we have    n;1 X p=l k +j(m k +1 +n k +1 ) '(fpx); ; n;lk;j(mk +1+nk+1)    2(n;lk;j(mk +1+nk+1)) jj'jjC 0 2(nk +1+mk+1) jj'jjC 0:

Collecting all estimates together one has    n;1 X p=0 '(fpx);n    Rk+ (lk+jnk +1)Var ; ' "_2k;2 + 2 nk+1+ (j+ 1)mk+1  jj'jjC0+jnk +1k+1:

Now, sincen > lk+j(nk+1+mk+1), andlk> Nk, we obtain    1 n n;1 X p=0 '(fp_x);   < R k lk +Var ; ' "_2k;2 +2n k+1+mk+1 Nk +mk +1 nk+1  jj'jjC0+k +1:

Since the right hand side tends to 0 ask!1, andk!1forn!1, we nally

conclude that lim n!1 1 n n;1 X p=0 '(fp_{x) =}

for allx2F, and hence,F K .

Proof of Lemma 5.4. We are going to show that for every continuous function 

there exist a limit

I() = lim_k

!1 Z

 dk: (29)

Obviously, ifI() is well dened, thenI is a positive linear functional onC(XR).

Hence by the Riesz theorem there exist a unique probability measureonX such that

I() =

Z

 d for every2C(XR)

and thus,k!weakly.

Let us prove (29). It is sucient to show that for every  > 0 there exists

K=K()>0 such that for allk1k2> Kone has    Z  dk1 ; Z  dk2   =    1 #(Lk1) X x2L k 1 (x); 1 #(Lk2) X y2L k 2 (y)  < :

Without loss of generality we may assume thatk1> k2. Then    1 #(Lk1) X x2L k 1 (x); 1 #(Lk2) X y2L k 2 (y)    1 #(Lk1) X x2L k 1   (x) ;(y(x))    wherey(x)2Lk

2 is a uniqie point in Lk2 such thatx descends fromy(x). Taking

into account the way the setsLk were constructed, we conclude that d(xy(x))

"

2k1:

Hence, fork1k2> K one has    Z  dk1 ; Z  dk2    sup  j(x);(y)j: d(xy)< " 2K  !0 asK!1:

Now, we have to show that (F) = 1. Note, that k+p(Fk) = 1 for all p 0,

sinceFk+p

Fk andk

+p(Fk+p) = 1 by construction. Sinceis the weak limit of fkg, and Fk are closed, using the properties of weak convergence of measures we

obtain

(Fk) lim

p!1

k+p(Fk) = 1

and hence(Fk) = 1. Finally, sinceF =T

kFk, one has (F) = 1.

Proof of Lemma 5.5. By the denition,Bn(x") is an open set, thus, sincek!,

we have (Bn(x")) lim k!1 k(Bn(x")) = lim k!1 1 #(Lk)#(fz2Lk: z2Bn(x")g): Supposenl

1=n1, then there existsk

1 such that

lk < nlk +1:

As in the proof of Lemma 5.3, letj2f0::: Nk +1

;1gbe such that

lk+ (nk+1+mk+1)j < n

lk+ (nk

+1+mk+1)(j+ 1):

We start by showing that #(Bn(x")\Lk)1, and thusk(Bn(x"))#(Lk) ;1.

Indeed, suppose there two points z1z2

2 Lk such that z 1z2

2 Bn(x") as

well. This means that dn(z1z2) < 2". However, from (19) we know that

dlk(z

1z2) > 5". Hence, we have arrived at contradiction, since n > lk and thus

dn(z1z2) dl

k(z 1z2).

We continue by showing thatk+1(

Bn(x")) does not exceed (#(Lk)Mj +1

k+1) ;1.

Suppose, two points z1z2 2 Lk

+1 are in

Bn(x") as well. Therefore, there exist

pointsx1x2 2Lk andy 1y2 2Dk +1 such that z1=z(x1y1) z2=z(x2y2):

All the points inDk+1 are obtained by shadowing certain combinatations of points

fromCk+1 (see (14)), i.e.,

y1=y(i1::: iN k +1) y 2=y(i 0 1::: i 0 Nk +1) where (i1::: iN k +1)(i 0 1::: i 0 Nk +1) 2f1::: Mk +1 gN k +1.

We claim that necessarily x1 = x2 and (i1::: ij) = (i 0 1::: i 0 j). Indeed, if x1 6 =x2 then dlk(x 1x2) dl k(x 1z1) +dl k(z 1x) +dl k(xz 2) +dl k(z 2x2)  " 2k +"+"+₂"k 5"

and thus we have a contradiction with (19). Similary we proceed with our second claim. If j = 0 there is nothing to prove. Suppose j > 0 and there exists t,

1tj, such thatit6=i 0 t. Sincey1=y(i1::: iN k +1), andy 2=y(i 0 1::: i 0 Nk +1), one has dnk +1(x k+1 it f aty 1)< " 2k+1 dn k +1(x k+1 i0 t faty 2)< " 2k+1: Moreover, dtk +1(z 1y1)< " 2k+1 dt k +1(z 2y2)< " 2k+1 and hence dnk +1(x k+1 it x k+1 i0 t ) dn k +1(x k+1 it f aty 1) +dt k +1(y 1f lk +m k +1z 1) + dn(z1z2) +dt k +1(f lk+m k +1z 2y2) +dn k +1(f aty 2xki 0 t)  " 2k+1 + " 2k+1 + 2"+ " 2k+1 + " 2k+1 <6"

which contradicts the fact that dnk +1(x

k+1 it x k+1 i0 t ) > 8", since xk+1 it x k+1 i0 t are dierent points in a (nk+18")-separated setCk+1.

Since (i1::: ij) is the same for all points z = z(xy(i1::: ij::: iN k +1))

which can lie inBn(x"), we easily conclude that there are at mostM

Nk +1 ;j k+1 such points. Hence k+1( Bn(x")) 1 #(Lk)MNk +1 k+1 MNk +1 ;j k+1 = 1 #(Lk)M_kj+1

For anyp >1 one has

k+p(

Bn(x"=2))

1 #(Lk)Mkj+1

as well. This is indeed the case, because the points ofLk+p, which lie in

Bn(x"=2),

can only descend from the points of Lk+1, which are in

Bn(x"). We provethis

nally by contradiction. Suppose we can nd pointsz1 2Lk

+1 and z2 2Lk

+p,z2

descends fromz1such that

dn(z2x)< "=2 and dn(z1x)> ":

This implies that dn(z1z2)

 dn(z 1x)

;dn(xz

2) > "=2. The latter however is

not possible, since

dn(z1z2) dl k +1(z 1z2)  " 2k+2 + " 2k+3 +:::= " 2k+1:

Hence there are exactly #(Dk+2):::#(Dk+p) points inLk+p,p

2, which descend

from a given point inLk+1. Hence

k+p( Bn(x"=2)) MNk +1 ;j k+1 #(Dk +2):::#(Dk+p) #(Lk)MNk +1 k+1 #(Dk +2):::#(Dk+p) = _#(L 1 k)Mkj+1 : And therefore (Bn(x"=2)) lim p!1 k+p( Bn(x"=2)) 1 #(Lk)M_kj+1 :

Now, by the choice ofkandj we have n;lk;j(nk +1+mk+1) nk +1+mk+1 wherelk =N1n1+N2(n2+m2) +:::+Nk(nk+mk). Therefore n;lk;j(nk +1+mk+1) lk+j(nk+1+mk+1)  nk+1+mk+1 Nk !0 ask!1

because of the choice of Nk. Since Mk has been chosen in a such way that Mk exp(snk), andmk are much smaller thannk, for largekwe obtain

#(Lk)Mkj+1=M N1 1 :::M Nk k Mkj+1 exp  s(N1n1+N2n2+:::+Nknk+jnk+1)  exp  (s; =2)(N 1n1+:::+Nk(nk+mk) +j(nk+1+mk+1)  exp ; (s; )n

Therefore, sincek!1asn!1, for all suciently largenone has

(Bn(x"=2))exp(;n(s; ))

for everyxsuch that Bn(x"=2)\F6=?.

Acknowledgements.

We are grateful to Ai Hua Fan for making available to us a copy of 7], and to Jorg Schmeling for valuable discussions. The second author acklowledges the support of The Netherlands Organization for Scientic Research (NWO), grant 613-06-551.

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