The recurrence dimension for piecewise monotonic maps of the interval

The recurrence dimension for piecewise monotonic maps of the interval

FRANZ HOFBAUER

Abstract. We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise mono- tonic transformation on the interval [0,1], giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and charac- teristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function.

Mathematics Subject Classification (2000):37E05 (primary); 37C45, 28A80, 37B40, 28A78 (secondary).

1.

Introduction

In the last years there was much interest in generalized notions of dimension. In [14] a theory of dimension structures is developed. Special cases of these structures are weighted versions of Hausdorff dimensions. We define them only on the interval [0,1]. Let|U|be the diameter of a subsetU of [0,1]. Forδ >0 aδ-cover of a set A⊂[0,1] is a finite or countable collection of intervals of length≤δcovering A.

For a set functionwwith values in

R

^+ands,t ∈

R

we define forA⊂[0,1]

νs,t(A)= lim

δ→0inf

C

U∈C

w(U)^s|U|^t

where the infimum is taken over all δ-covers

C

^of A. It follows from the general theory in [14] that for fixeds ∈

R

there is a critical valuet_c ∈[−∞,∞] such that νs,t(A)= ∞fort <t_c andνs,t(A)=0 fort >t_c. We denote this critical valuet_c byd_s(A)and call it the dimension of the setA.

Ifmis a probability measure on [0,1] and if one setsw(U)=m(U), one gets the dimension introduced by Olsen in [13] and called multifractal generalization of Research for this paper was partially carried out while the author was visiting the Max Planck Institut for Mathematics in Bonn as a participant in the activity on Algebraic and Topological Dynamics.

Pervenuto alla Redazione il 3 febbraio 2005 e in forma definitiva il 2 agosto 2005.

the Hausdorff dimension there. For invariant sets of piecewise monotonic maps on the interval, this dimension is investigated in [9].

If T : [0,1] → [0,1] is a map, one can define the recurrence time of a set U ⊂[0,1] byτ(U)=min{n ≥1 :Tⁿ(U)∩U = ∅}. If one setsw(U)=e^−τ(U) one gets the dimension introduced in [1] (see also [2]) and called the spectrum of dimensions for Poincar´e recurrences there. This is the dimension we consider in this paper. We call it recurrence dimension for simplicity.

For topological Markov chains and conformal repellers satisfying a specifica- tion property this recurrence dimension has been computed in terms of topologi- cal entropy or implicitly using a pressure function (see [2]). In this paper we do the same for invariant subsets of piecewise monotonic maps on the interval [0,1].

We say that T : [0,1] → [0,1] is piecewise monotonic if there are numbers 0= c₀ <c₁ < · · ·< c_N = 1, such thatT|(c_i−1,c_i)is monotone and continuous for 1≤i ≤ N. Set

Z

= {(c₀,c₁), . . . , (c_N−1,c_N)}andP = {c₀,c₁, . . . ,c_N}. We assume that the dynamical system([0,1],T)has only finitely many or no attracting periodic orbits. In chapter IV of [12] it shown that this holds forC²-mapsT with non-flat critical points. In the second part of this paper we assume thatT is expand- ing, which implies that there can be no attracting periodic orbits. Furthermore, we assume thatT has a continuous derivativeTon [0,1]\Pand setϕ= −log|T|. We shall need regularity conditions forϕ. To this end letC be the set of all func- tions f : [0,1]\P→

R

, such that for each Z ∈

Z

^{the map f}|Z can be extended to a continuous real-valued function on the closure Z of Z. Furthermore let D be the set of all functions f : [0,1]\P→

R

, such that for each Z∈

Z

^{the map f}|Z can be extended to a continuous function f_Z : Z →(−∞,∞]. For p ≥1 letD^p be the set of all f ∈ Dsuch thate^−f is of bounded p-variation. The p-variation of a functiong: [0,1]→

R

is defined by var^pg:=supn

j=1|g(x_j)−g(x_j−1)|^p where the sup is taken over n ≥ 1 and 0 ≤ x₀ < x₁ < · · · < x_n ≤ 1. Notice thatϕ ∈ C excludes the possibility thatT attains the value zero, whereas this is possible on P, ifϕ∈D.

In the first part of the paper we investigate the dimension of an invariant mea- sure. For a probability measureµwe define

d_s(µ)=inf{d_s(A):µ(A)=1}.

Theorem 2.4 below gives a local result. If ϕ is either in C or in D^p for some p ≥ 1 and if µ is an ergodic T–invariant measure with h_µ > 0, then χ_µ :=

−µ(ϕ)is greater than zero and lim_r→0^τ(₋^B_log^r(x_r⁾⁾ = _χ¹_µ holds forµ-almost all x. In this paper we prove only lim sup_r→0^τ(₋^B_log^r(x_r⁾⁾ ≤ _χ¹_µ using methods similar to those used in [7] and [8] for Hausdorff dimension and local dimension of an ergodic invariant measure. The other part of this theorem, namely lim inf_r→0^τ(₋^B_log^r(x_r⁾⁾ ≥

χ1µ, is already shown in [15]. Using Theorem 2.4 and results from [7] on local dimension we get then thatd_s(µ) = ^hµ_χ⁻_µ^s for everys ∈[0,h_µ). This is stated as

Theorem 2.6 below. Similar results are known for subshifts with positive entropy satisfying a specification property (see [3]).

In the second part of this paper we investigate the dimension of a closed subset Aof [0,1], which is invariant under the piecewise monotonic mapT. We assume thatT is expanding, which means that inf|T|> 1, and thatϕ ∈C. Furthermore, we set

Z

n=_n−1

j=0T^−j

Z

. For a function f ∈C we define p(T|A, f)=lim sup

n→∞

1

nlog

Z∈Zn,Z∩A=∅

sup

x∈Z∩A

e

_n−1 j=0f(T^jx).

This is the definition of pressure for the isomorphic shift space one gets by coding with respect to the partition

Z

. Since the transformationT can have discontinuities on the set P, we use p(T|A, f)as the notion of pressure. Setπ(t)= p(T|A,tϕ) for t ≥ 0. For s ∈ [0,h_top(T|A)) we show then that there is a unique solution z_s(A) >0 of the equationπ(t) = s, which equalsd_s(A). This is stated as Theo- rem 3.1 below.

2.

Dimension of an invariant measure

In order to investigate the recurrence dimension of an ergodic invariant measure µwithh_µ > 0 we use Markov extensions introduced in [5] and [10] and used in [7] and in [8] for the investigation of Hausdorff dimensions. Let

Y

be a finite or countable collection of open pairwise disjoint intervals, such thatT|Y is monotone for allY ∈

Y

and such thatµ(

Y∈YY)=1. SetR=_∞

j=0T^−j(

Y∈YY). Since µis invariant, we haveµ(R) = 1. Forn ≥ 1 letY_n(x)be the unique element of

Y

n =_n−1

j=0T^−j

Y

, which containsx. Ifx ∈ R, thenY_n(x)exists for alln≥1.

The Markov extension of the dynamical system([0,1], µ,T)with respect to

Y

is constructed as follows. If D is a subinterval of some Y ∈

Y

, we call the nonempty sets amongT(D)∩Y forY ∈

Y

the successors ofD. We writeD→C, ifC is a successor of D. The successors are again subintervals of elements of

Y

^, so we can iterate the formation of successors. Set

D

1=

Y

^and

D

k+1 =

D

k ∪ {C : there isD ∈

D

k withD→C}fork≥1. Finally set

D

=_∞

k=1

D

k.

In order to get a dynamical system, take disjoint copies Dˆ of the intervals D ∈

D

^{and set Xˆ} =

D∈DD. Letˆ q : Dˆ → [0,1] be the imbedding, such that q : Xˆ → [0,1] is defined. If D ∈

D

^andx ∈ D∩

Y∈YY thenT(x)is defined and contained in

D→CC. For the corresponding pointxˆ ∈ ˆDwe letTˆ(xˆ)be the point in

D→CCˆ corresponding toT(x). In this way we have definedTˆ on the subsetq⁻¹(

Y∈YY)ofX, such thatˆ q◦ ˆT =T◦qholds. Note thatq⁻¹(

Y∈YY) containsq⁻¹(R). IfH_µ(

Y

)= −

Y∈Yµ(Y)logµ(Y) <∞, it is shown in [8] (for finite

Y

in [11]) that there is a probability measureµˆ on Xˆ withµ(ˆ q⁻¹(R)) = 1,

such thatTˆ is definedµˆ-almost everywhere and has the following properties ˆ

µ is aTˆ-invariant ergodic measure, ˆ

µ(q⁻¹(A))=µ(A) for all Borel subsetsAof [0,1].

Set

Y

ˆ = { ˆD: D∈

D

}. By the above construction,

Y

ˆ is a finite or countable Markov partition of the dynamical system (Xˆ,Tˆ,µ)ˆ . This explains the name Markov ex- tension. Forn≥1 letYˆ_n(xˆ)be that element of

Y

ˆn =_n−1

j=0Tˆ^−j

Y

ˆ, which contains ˆ

x. Ifxˆ∈q⁻¹(R)thenYˆ_n(xˆ)exists for alln≥1.

We investigate recurrence on the Markov extension.

Lemma 2.1. There is a setMˆ ⊂ ˆX withµ(ˆ Mˆ)=1such thatlim sup

n→∞

1

nτ(Yˆ_n(xˆ))≤1 for allxˆ ∈ ˆM.

Proof. Choose D ∈

D

^with µ(ˆ Dˆ) > 0. By the ergodic theorem there is a set Mˆ_D ⊂ ˆX withµ(ˆ Mˆ_D)= 1 such that lim_n→∞ ¹_nn−1

j=01_Dˆ(Tˆ^j(xˆ))= ˆµ(Dˆ)holds for allxˆ ∈ ˆM_D.

Letxˆ ∈ ˆD∩ ˆM_Dand letn₀=0<n₁<n₂< . . . be those integersn, for which Tˆⁿ(xˆ)∈ ˆD holds. The above result says that lim_j→∞n^j

j = ˆµ(Dˆ) > 0. Since

Y

ˆ is a Markov partition, we have either Dˆ ⊂ ˆT^nj(Yˆ_nj(xˆ))or Dˆ ∩ ˆT^nj(Yˆ_nj(xˆ))= ∅. SinceTˆ^nj(xˆ) ∈ ˆD∩ ˆT^nj(Yˆ_nj(xˆ))we conclude that Dˆ ⊂ ˆT^nj(Yˆ_nj(xˆ))for j ≥ 1.

For arbitraryn ≥ 1 choose j such thatn_j−1 < n ≤ n_j. Then Yˆ_nj(xˆ) ⊂ ˆY_n(xˆ) andxˆ ∈ ˆD ⊂ ˆT^nj(Yˆ_n(xˆ))follows. Sincexˆ ∈ ˆY_n(xˆ)we getτ(Yˆ_n(xˆ)) ≤ n_j and

1

nτ(Yˆ_n(xˆ))≤ _nⁿ_j−1^j . Furthermore, lim_{j→∞ n}^nj

j−1 = lim_{j→∞ j−}^j₁ⁿ_j^j_n^j−1

j−1 = 1. This implies lim sup_n→∞ ¹_nτ(Yˆ_n(xˆ))≤1.

Set

D

^∗= {D∈

D

^:µ(ˆ Dˆ) >0}andMˆ =

D∈D^∗(Dˆ ∩ ˆM_D). We have shown, that lim sup_n→∞n¹τ(Yˆ_n(xˆ))≤1 holds for allxˆ ∈ ˆM. Sinceµ(ˆ Dˆ ∩ ˆM_D)= ˆµ(Dˆ) we get

ˆ

µ(Mˆ)=

D∈D^∗

ˆ

µ(Dˆ ∩ ˆM_D)=

D∈D^∗

ˆ

µ(Dˆ)=

D∈D

ˆ

µ(Dˆ)= ˆµ(Xˆ)=1 and the proof is finished.

We bring down this result to the original dynamical system.

Lemma 2.2. We havelim sup_n→∞n¹τ(Y_n(x))≤1forµ-almost all x ∈[0,1].

Proof. Let Mˆ be as in Lemma 2.1 and set M = q(Mˆ)∩ R. Then q⁻¹(M) ⊃ Mˆ ∩q⁻¹(R)and henceµ(M)= ˆµ(q⁻¹(M))=1.

Forx ∈ M there isxˆ ∈ ˆM withq(xˆ)= x. Suppose thatD_n ∈

D

is such that Tˆⁿ(xˆ)∈ ˆD_nfor alln≥0. We haveq(Dˆ_n)= D_n. By the definition of successor and

because ofq◦ ˆT =T◦qthere areU₀,U₁,· · · ∈

Y

, such thatTⁿ(x)∈ D_n ⊂U_nfor n ≥0. It follows thatY_n(x)= _n−1

j=0T^−jU_j andYˆ_n(xˆ)=_n−1

j=0Tˆ^−jDˆ_j. We get q(Yˆ_n(xˆ))=_n−1

j=0T^−jD_j ⊂Y_n(x). Ifτ(Yˆ_n(xˆ))=k, thenTˆ^k(Yˆ_n(xˆ))∩ ˆY_n(xˆ)= ∅, and sinceq◦ ˆT^k =T^k◦qholds we getT^k(Y_n(x))∩Y_n(x)= ∅. Henceτ(Y_n(x))≤ τ(Yˆ_n(xˆ)). It follows from Lemma 2.1 that lim sup_n→∞ ¹_nτ(Y_n(x))≤1 holds for all x∈ M, which means forµ-almost allx.

Lemma 2.3. Let T be a piecewise monotonic transformation, which has only finitely many or no attracting periodic orbits. Suppose thatϕ= −log|T|is either in C or in D^p for some p ≥ 1. Letµ be an ergodic T –invariant measure with h_µ>0. Thenµ(ϕ)≤0andϕ∈L¹(µ).

Proof. Suppose thatµ(ϕ) >0 holds. There is a continuous functionψ : [0,1]→

R

^satisfyingψ ≤ ϕ andµ(ψ) > 0. Let (Xˆ,Tˆ,µ)ˆ be the Markov extension of ([0,1],T, µ)with respect to the partition

Z

^{and letq}be the corresponding projec- tion. By the ergodic theorem, there isxˆ ∈ ˆX, such that ¹_nn−1

j=0δ_Tˆj(xˆ) converges weakly toµˆ. Sinceµˆ is ergodic and has nonzero entropy, the set of ally, for whichˆ _∞

j=1Yˆ_n(yˆ)is a nondegenerate interval, has µˆ-measure zero, and hence we can assume that_∞

j=1Yˆ_n(xˆ) = { ˆx}. There is a sequence D₀D₁D₂. . . in

D

^{such that} Tˆ^k(xˆ)∈ ˆD_kfork≥0. Since

Y

ˆ is a Markov partition, we haveTˆ(Dˆ_k)⊃ ˆD_k+1for k ≥0. There isD ∈

D

^withµ(ˆ Dˆ) >0, and hence Doccurs infinitely often in this sequence. Choosen₁ < n₂ <n₃ < . . . such thatD_nk = D. For everym there is a point yˆ of periodn_m−n₁withTˆ^j(yˆ)∈ ˆD_n1+j for 0≤ j ≤n_m−n₁. Letνˆm be the periodic-orbit-measure supported by the periodic orbit ofy. Then the sequenceˆ

ˆ

νm converges weakly toµˆ. Since q : Xˆ → [0,1] is continuous, the measures νm = ˆνm◦q⁻¹are periodic-orbit-measures on([0,1],T), which converge weakly toµ. Sinceψ is continuous, there ism₀withνm(ψ) >0 form ≥m₀. Because of ϕ ≥ψwe have alsoνm(ϕ) >0 form ≥m₀. This implies, that the periodic orbits supporting the measuresνm form ≥ m₀are attracting, contradicting the assump- tion, that at most finitely many attracting periodic orbits exist. This contradiction shows thatµ(ϕ)≤0. Since we assume thatϕis inC or inD^p, it follows thatϕis bounded below and hence inL¹(µ).

Theorem 2.4. Let T be a piecewise monotonic transformation, which has only finitely many or no attracting periodic orbits. Suppose thatϕ= −log|T|is either in C or in D^p for some p ≥ 1. Letµ be an ergodic T –invariant measure with h_µ > 0. Thenχ_µ := −µ(ϕ)is greater than zero andlim_r→0^τ(₋^B_log^r(x_r⁾⁾ = _χ¹_µ holds forµ-almost all x.

Proof. By Lemma 2.3 we haveϕ ∈ L¹(µ)and by Theorem 1 in [6] we getχ_µ =

−µ(ϕ) >0. Chooseε∈(0,^χ₂^µ). By Lemma 1 in [8] there is a finite or countable

collection

Y

of pairwise disjoint intervals, such that µ

Y∈Y

Y

=1;

sup

x∈Y

ϕ(x)− inf

x∈Yϕ(x) < εfor everyY ∈

Y

; H_µ(

Y

)= −

Y∈Y

µ(Y)logµ(Y) <∞.

Now let M_ε be a set ofµ-measure one, such that lim sup_n→∞ ¹_nτ(Y_n(x))≤1 and that lim_n→∞¹_nn−1

j=0ϕ(T^jx) = µ(ϕ) = −χ_µ for all x ∈ M_ε. This is possible because of Lemma 2.2 and the ergodic theorem. For x ∈ M_ε we get by the mean value theorem and asTⁿ(Y_n(x))⊂[0,1] that

log|Y_n(x)| ≤ sup

y∈Y_n(x) n−1

j=0

ϕ(T^jy)+log|Tⁿ(Y_n(x))| ≤ sup

y∈Y_n(x) n−1

j=0

ϕ(T^jy).

Since sup_Uϕ−inf_Uϕ <εfor everyU∈

Y

^{we getn}_j=⁻¹₀ϕ(T^jy)≤_n−1

j=0ϕ(T^jx)+

nε, if y∈Y_n(x), and hence log|Y_n(x)| ≤_n−1

j=0ϕ(T^jx)+nε. It follows that

nlim→∞

1

nlog|Y_n(x)| ≤ −χ_µ+ε . In particular we have lim_n→∞|Y_n(x)| =0, as−χ_µ+ε <0.

For x ∈ M_ε andr > 0 let n be such that |Y_n(x)| ≤ r < |Y_n−1(x)|. This impliesY_n(x)⊆ B_r(x), and therefore

τ(B_r(x))

−logr ≤ τ(Y_n(x))

−log|Y_n−1(x)| =

1

nτ(Y_n(x))

−¹_nlog|Y_n−1(x)|.

There is n₀, such that ¹_nτ(Y_n(x)) < 1+ε and−¹_nlog|Y_n−1(x)| > χ_µ −2ε for n≥n₀. Forr <|Y_n0(x)|we get ^τ(₋^B_log^r(x_r⁾⁾ < _χ¹_µ^+ε_−2ε and hence also

lim sup

r→0

τ(B_r(x))

−logr ≤ 1+ε χ_µ−2ε. Now setM =_∞

k=1M_1/k. Then we haveµ(M)=1 and lim sup_r→0^τ(₋^B_log^r(x_r⁾⁾ ≤ _χ¹_µ for allx∈ M, which means forµ-almost allx.

Theorem 2 in [15] says that lim inf_r→0^τ(₋^B_log^r(x_r⁾⁾ ≥ _χ¹_µ holds for µ-almost all x. It is proved there first for cylinder sets with respect to a partition for general dynamical systems and then an approximation procedure as above is used.

In the proof of the result on the dimension of a measure we shall use centered covers. Forε > 0 we call

F

^{a centered} δ–cover of A, if A ⊆

C∈FC and for everyC ∈

F

there exists anx ∈Aand anα∈(0, δ] withC =(x−α,x+α). The next lemma shows that we can use centered covers instead of covers.

Lemma 2.5. For A ⊂[0,1]letν˜s,t(A)be defined in the same way asνs,t(A)but with centered covers instead of covers. Then νs,t(A) ≤ ˜νs,t(A) ≤ 2νs,t(A)for s,t ≥0.

Proof. Since a centeredδ-cover is also aδ-cover, it follows thatνs,t(A)≤ ˜νs,t(A). Let I be an interval of length≤δwith I ∩A= ∅. LetI₁be the left half and I₂the right half of I. If I₁∩ A= ∅, there isx₁ ∈ I₁∩ AwithI₁⊂ B_r1(x₁)⊂ I, where r₁ is the distance from x₁ to the left endpoint of I. If I₁ ∩ A = ∅, set B_r1(x₁)= ∅. Similarly, ifI₂∩A= ∅, there isx₂ ∈I₂∩ AwithI₂⊂ B_r2(x₂)⊂ I, where r₂ is the distance from x₂ to the right endpoint of I. If I₂∩ A = ∅, set B_r2(x₂)= ∅. We haveA∩(B_r1(x₁)∪B_r2(x₂))= A∩I. If

C

^{is a}δ-cover ofA, for eachI ∈

C

^{choose B}r₁(x₁)andB_r2(x₂)as above. The nonempty sets among them form a centeredδ-cover

C

˜^{ofAsuch that}_{C˜∈ ˜C}^e−sτ(C˜)| ˜C|^t ≤2

C∈Ce^−sτ(C)|C|^t, sinceU ⊂ V impliesτ(U) ≥ τ(V)and|U| ≤ |V|. From this inequality we get

˜

νs,t(A)≤2νs,t(A).

Theorem 2.6. Let T be a piecewise monotonic transformation, which has only finitely many or no attracting periodic orbits. Suppose thatϕ= −log|T|is either in C or in D^p for some p ≥ 1. Letµ be an ergodic T –invariant measure with h_µ>0. Then for every s∈[0,h_µ)we have d_s(µ)= ^hµ_χ⁻_µ^s, whereχ_µ= −µ(ϕ). Proof. Setα=^h_χ^µ_µ andβ=_χ¹_µ. By Theorem 2.4 we haveχ_µ>0 and lim

r→0 τ(Br(x))

−logr = β forµ-almost allx. By Theorem 1 in [7] we have lim_r→0^logµ(_log^B_r^r(x)) = αforµ- almost allx.

In order to prove d_s(µ) ≤ α−sβ choose an arbitrary t > α −sβ and set ε= ¹₂(t −α+sβ). Sinces <h_µ, we havet >0. Fork∈

N

^define

M_k={x∈[0,1] :µ(B_r(x))^t−εα ≥¹_kr^t andµ(B_r(x))^sβ−εα ≥¹_ke^{−sτ(Br(x))}for allr∈(0,1)}

Forµ-almost allx∈[0,1] there exists anr₀(x) >0 such that t−ε

α

logµ(B_r(x))

logr ≤t and sβ−ε

α

−logµ(B_r(x))

τ(B_r(x)) = sβ−ε α

logµ(B_r(x)) logr

−logr τ(B_r(x)) ≤s

for allr ∈(0,r₀(x)). These are the inequalities in the definition ofM_k withk=1.

Hence forµ-almost all x ∈ [0,1] there is ak ∈

N

^{with x} ∈ M_k, and therefore µ(_∞

k=1M_k)=1.

Fixk ∈

N

^andδ∈ (0,1), and let

C

be a centeredδ–cover ofM_k. Since each C ∈

C

is a ball with center inM_k we get by the definition ofM_k

C∈C

e^−sτ(C)|C|^t ≤2^tk²

C∈C

µ(C)^sβ−εα µ(C)^t−εα =2^tk²

C∈C

µ(C)≤2^t+1k². The last inequality follows, since we can assume that each x ∈[0,1] is contained in at most two element of

C

. Otherwise we can cancel elements from

C

^{and have} still a cover. This implies that ν˜s,t(M_k) < 2^t+1k². By Lemma 2.5 we get also νs,t(M_k) < 2^t+1k². By the definition of dimension we haved_s(M_k) ≤ t. For u > t we have thereforeνs,u(M_k) = 0 for allk, and asνs,u is an outer measure, we getνs,u(_∞

k=1M_k)=0. It follows thatd_s(_∞

k=1M_k)≤t and henced_s(µ)≤t, since µ(_∞

k=1M_k) = 1. As t can be chosen arbitrarily close toα−sβ, we get d_s(µ)≤α−sβ.

In order to prove thatα−sβ ≤d_s(µ), it suffices to showµ(L)=0 for every Borel setL ⊆[0,1] withd_s(L) < α−sβ. LetLbe such a set. Sinces <h_µ, we haveα−sβ >0. Chooset >0 withd_s(L) <t < α−sβand setε= ¹₂(α−sβ−t). Fork∈

N

^define

L_k={x∈L :µ(B_r(x))^t+εα ≤kr^t andµ(B_r(x))^sβ+εα ≤ke^{−sτ(Br(x))}for allr∈(0,1)}

Forµ-almost allx∈Lthere is anr₀(x) >0 such that t+ε

α

logµ(B_r(x))

logr ≥t and sβ+ε

α

−logµ(B_r(x))

τ(B_r(x)) = sβ+ε α

logµ(B_r(x)) logr

−logr τ(B_r(x)) ≥s

holds for allr ∈(0,r₀(x)). These are the inequalities in the definition ofL_k with k = 1. Hence forµ-almost allx ∈L there is ak ∈

N

^withx ∈ L_k, and therefore µ(L)=µ(_∞

k=1L_k). It remains to showµ(L_k)=0 for allk∈

N

^.

To this end fix k ∈

N

^and η > 0. Since d_s(L_k) ≤ d_s(L) < t we have νs,t(L_k)=0 and hence alsoν˜s,t(L_k)=0 by Lemma 2.5. There exists aδ∈(0,1) and a centeredδ–cover

C

^ofLkwith

C∈C

e^−sτ(C)|C|^t < 2^t k²η .

Since eachC ∈

C

is a ball with center inL_k we get by the definition ofL_k that µ(L_k)≤

C∈C

µ(C)=

C∈C

µ(C)^sβ+εα µ(C)^t+εα ≤ k² 2^t

C∈C

e^−sτ(C)|C|^t < η . Asη >0 was arbitrary we getµ(L_k)=0, which completes the proof.

Remark 2.7. The statement in Theorem 2.6 holds also fors < 0. This can be shown by the method developed in chapter 5 of [16], which is based on a result in [4]. But it seems that the proof fors >0 cannot be given by this method. One gets only an inequality in this case.

3.

Dimension of an invariant set

In order to investigate the dimension of a closed invariant set Ain the dynamical system([0,1],T), we assume that T is expanding, this means that inf|T| > 1.

Furthermore, we assume thatT ∈C. This implies thatϕ= −log|T|is also inC.

For a function f ∈Cwe define p(T|A, f)=lim sup

n→∞

1

nlog

Z∈Zn(A)

sup

x∈Z∩A

e^Snf(x)

where S_nf(x) = _n−1

j=0 f(T^jx) and

Z

n(A) = {Z ∈

Z

n : Z ∩ A = ∅}. If we consider the dynamical system([0,1],T)as a shift space, this is the usual definition of pressure. In particular, the pressure does not change, if we refine

Z

^{, and the} variational principle holds. These and other properties of the pressure can be found in [17]. We define the functionπ : [0,∞)→

R

^by

π(t)= p(T|A,tϕ) .

For t₁ < t₂ we have t₂ϕ ≤ t₁ϕ +(t₂− t₁)supϕ and hence π(t₂) ≤ π(t₁)+ (t₂ −t₁)supϕ. Since supϕ < 0 this implies that π is strictly decreasing and that lim_t→∞π(t) = −∞. Furthermore, π(0) = h_top(T|A) ≥ 0. Hence for s ∈ [0,h_top(T|A)] there is a unique t˜ ≥ 0 withπ(t˜) = s, with π(t) > s for t < t˜and withπ(t) < s fort > t˜. We denote thist˜byz_s(A). Ifs < h_top(T|A) thenz_s(A) >0.

Theorem 3.1. Let T be an expanding piecewise monotonic transformation on[0,1], such that T ∈ C, and A a closed invariant subset with h_top(T|A) > 0. For s ∈[0,h_top(T|A))we have then d_s(A)=z_s(A).

Proof. Sinces < h_top(T|A)we havez_s(A) > 0. Chooset ∈ (0,z_s(A)). Since t < z_s(A), we get p(T|A,tϕ) = π(t) > s. By the variational principle there is an ergodic invariant measure µon Asatisfying h_µ +tµ(ϕ) > s. Because of µ(ϕ) ≤supϕ < 0 we geth_µ > s. Sinceϕ ∈ C we can apply Theorem 2.6 and get d_s(µ) = ^hµ_χ⁻_µ^s. Since h_µ −s > −tµ(ϕ) = tχ_µ we get d_s(µ) > t. This implies d_s(A) > t. As t can be chosen arbitrarily close to z_s(A), the inequality d_s(A)≥z_s(A)is shown.

Choose t > z_s(A). Then t > 0 and π(t) < s. Fix ε ∈ (0,^s−π(_t ^t)). We choose points 0 = d₀ < d₁ < · · · < d_K = 1 containingc₀,c₁, . . . ,c_N and set

Z

= {(d₀,d₁), . . . , (d_K−1,d_K)}, such thatT|Z is monotone and sup_x∈Zϕ(x)− inf_x∈Zϕ(x) < ε for every Z ∈

Z

. If we define the pressure with respect to this finer partition

Z

we get the same as for the original

Z

^{. Set}α= ^s−π(₂^t)−εt >0 and

p_n =e^n(tε−s)

Z∈Zn(A)sup_Z∩Ae^{t Snϕ}forn≥1. Then lim sup

n→∞

1

nlogp_n =tε−s+lim sup

n→∞

1

nlog

Z∈Zn(A)

sup

Z∩A

e^{t Snϕ} =tε−s+π(t)= −2α . Hence there is a constantc>0, such that p_n≤ce^−nαholds for alln≥1.

Set

Z

_n^k(A)= {Z ∈

Z

n(A):τ(Z)=k}and

Z

_nⁿ(A)= {Z ∈

Z

n(A): τ(Z)≥ n}. Suppose thatZ =_n−1

j=0T^−jZ_j ∈

Z

_n^k(A), where Z₀,Z₁, . . . ,Z_n−1 ∈

Z

^and k < n. Then T^k(Z)∩ Z = ∅ and hence Z_j+k = Z_j for 0 ≤ j ≤ n−k −1.

Set Z˜ = _k−1

j=0T^−jZ_j. ThenZ ⊂ ˜Z and Z˜ ∈

Z

k(A), as A∩ Z = ∅and hence A∩ ˜Z= ∅. Furthermore, the mapZ→ ˜Zfrom

Z

n^k(A)to

Z

k(A)is injective, since Z_k, . . . ,Z_n−1are determined byZ₁, . . . ,Z_k−1. Fork =nwe setZ˜ = Zand these results are trivial.

For each Z∈

Z

k(A)we get by the mean value theorem that

|Z| ≤ |T^kZ|sup

x∈Z

|T^(k)(x)| −1= |T^kZ|sup

x∈Z

e^Skϕ(x).

Since T^kZ ⊂ [0,1] we have|T^kZ| ≤ 1. Since sup_Uϕ−inf_Uϕ < ε for every U ∈

Z

^{we haveS}kϕ(x)−S_kϕ(y)≤kεwheneverxandyare in Z. As A∩Z = ∅ it follows that

|Z| ≤ sup

x∈A∩Z

e^Skϕ(x)+kε.

If Z ∈

Z

n^k(A)andZ˜ as above, we have|Z| ≤ | ˜Z|since Z⊂ ˜Z. Hence we get

Z∈Zn(A)

e^−sτ(Z)|Z|^t ≤ n k=1

e^−sk

Z∈Zn^k(A)

|Z|^t ≤ n k=1

e^−sk

˜ Z∈Zk(A)

| ˜Z|^t

≤ n

k=1

e^−sk

Z˜∈Zk(A)

sup

A∩ ˜Z

e^{t Skϕ+tkε}

=ⁿ

k=1

p_k ≤^∞

k=1

p_k ≤c ∞ k=1

e^−kα ≤ ce^−α 1−e^−α

Now

Z

n(A)covers the setAexcept for possibly finitely many points. We can cover these finitely many points by intervalsI₁,I₂, . . . ,I_j of arbitrary small length, such thatj

i=1e^−sτ(Ii)|I_i|^t <1 holds. SinceT is expanding, for everyδ >0 there isn such that the diameter of all intervals in

Z

n(A)is less thanδ. Hence for everyδ >0 we have found aδ-cover

C

^ofAwithC∈Ce^−sτ(C)|C|^t < ₁^ce₋^−α_e−α+1. This implies thatνs,t(A)≤ ₁^ce₋^−α_e−α +1 andd_s(A)≤t follows. We can choosetarbitrarily close toz_s(A)so thatd_s(A)≤z_s(A)is shown.

References

[1] V. AFRAIMOVICH,Pesins dimension for Poincar´e recurrences, Chaos7(1997), 12–20.

[2] V. AFRAIMOVICHand J. URIAS,Dimension–like characteristics of invariant sets in dy- namical systems, In: “Dynamics and randomness”, A. Maass, S. Martinez and J. San Martin (eds.), Kluwer Academic Publishers, Dordrecht, 2002, pp. 1–30.