Irregular sets of two-sided Birkhoff averages and hyperbolic sets

c 2015 by Institut Mittag-Leffler. All rights reserved

Irregular sets of two-sided Birkhoff averages and hyperbolic sets

Luis Barreira, Jinjun Li and Claudia Valls

Abstract. For two-sided topological Markov chains, we show that the set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the set of points for which the Birkhoff averages have a given set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the set of points for which the local entropies of an invariant measure on a locally maximal hyperbolic set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure set.

1. Introduction

We show that the irregular set of the points for which the two-sided Birkhoff averages of a given continuous function diverge is often residual. More precisely, for topologically mixing topological Markov chains we show that the irregular set is either empty or residual, even though in the context of ergodic theory it has zero measure with respect to any invariant measure.

Let f: X→X be a homeomorphism on a compact metric space. Given a continuous functionϕ:X→R, we consider the irregular set

X_ϕ=

x∈X: lim inf

n→∞

1 2n

n i=−n

ϕ fⁱ(x)

<lim sup

n→∞

1 2n

n i=−n

ϕ

fⁱ(x) .

It follows from Birkhoff’s ergodic theorem thatX_ϕ has zero measure with respect to any finitef-invariant measure onX. It turns out that from the topological point of view quite the contrary happens; namely, Xϕ is typically as large as the whole space. The following statement illustrates well this phenomenon. It is combination of our results with work of Barreira and Schmeling in [4].

Theorem 1.1. Let f: X→X be a topologically mixing topological Markov chain. Then for a C⁰-dense set of continuous functions ϕ:X→R the set Xϕ is residual.

More precisely, we show in the present paper that for each continuous func- tion ϕ the set Xϕ is either empty or residual (see Theorem 2.4). On the other hand, it is shown in [4] that there exists a C⁰-dense set S of continuous functions for which the irregular set is nonempty. For example,Scan be taken to be the class of (H¨older) continuous functions ϕsuch that:

1. ϕis a linear combination of characteristic functions of cylinder sets;

2. ϕis not cohomologous to a constant.

We recall thatϕis said to be cohomologous to a constant if there exist a bounded functionψ:X→Rand a constantcsuch that

ϕ=ψ−ψ◦f+c onX.

Clearly, ifϕis cohomologous to a constant, thenX_ϕ is the empty set.

More generally, we consider the following refined version of the irregular set.

Given an intervalI⊂R, let

XI=

x∈X:Aϕ(x) =I ,

whereA_ϕ(x) is the set of accumulation points of the sequence Sϕ(x, n) = 1

2n n i=−n

ϕ fⁱ(x)

.

Again for topologically mixing topological Markov chains, we show that when I is not a singleton andϕis an arbitrary continuous function, the setXI is either empty or residual (see Theorem2.1). This is our main result. Roughly speaking, the proof consists of bridging together strings of sufficiently large length corresponding to limits of two-sided Birkhoff averages. We note thatXI is a subset of the irregular setXϕwhenIis not a singleton and so Theorem1.1is in fact a consequence of the corresponding result for the setsX_I.

As an application, we obtain corresponding statements for the averagesSϕ(x, n) when X is a locally maximal hyperbolic set (see Theorem3.1). Here we describe only a nontrivial application of Theorem3.1to the local entropies.

We recall that if μ is a Gibbs measure of a continuous function ψ, assumed without loss of generality to have zero topological pressure, then the limit

(1) h_μ(x) := lim

n→∞− 1 2nlogμ

B_n(x, ε)

= lim

n→∞S_−ψ(x, n)

exists forμ-almost everyx∈X, where B_n(x, ε) =

n

k=−n

f^−kB

f^k(x), ε

and εis any sufficiently small constant. This is an immediate consequence of the invariance of the measure μ and the Shannon–McMillan–Breiman theorem (that usually is formulated for one-sided iterates). The numberhμ(x) (when defined) is called the local entropy ofμat the pointx(with respect tof). For the definitions and basic results of the theory we refer the reader to [2, Chapters 1–2].

The following result is a consequence of Theorem 3.1and identity (1).

Theorem 1.2. If μ is a Gibbs measure on a locally maximal hyperbolic set, then the set of points for which the local entropies do not exist, that is,the set

x∈X: lim inf

n→∞ − 1 2nlogμ

Bn(x, ε)

<lim sup

n→∞ − 1 2nlogμ

Bn(x, ε) ,

is either empty or residual.

We note that the irregular sets can also be very large from the point of view of dimension theory. In particular, it was shown in [4] that for a locally maximal hyperbolic setX of aC^1+εmapf that is topologically mixing and conformal onX (this means that the derivative off is a multiple of an isometry at each point ofX), ifϕis H¨older continuous and is not cohomologous to a constant, then the one-sided irregular set

Yϕ=

x∈X: lim inf

n→∞

1 n

n−1

i=0

ϕ fⁱ(x)

<lim sup

n→∞

1 n

n−1 i=0

ϕ

fⁱ(x)

of the setXϕ is as large as the whole space from the points of view of topological entropy and Hausdorff dimension, that is,

(2) h(f|Yϕ) =h(f|X) and dimHYϕ= dimHX,

where h(f|Z) is the topological entropy off on the set Z⊂X and dimHZ is the Hausdorff dimension of Z. Since the proof is based on the construction of nonin- variant measures obtained from concatenating Gibbs measures, the same argument applies with minor modifications to show that

h(f|X_ϕ) =h(f|X) and dimHXϕ= dimHX

for the two-sided irregular set. Now let Y=

ϕYϕ be the union over all H¨older continuous functions. Under the same hypotheses, we have

(3) h(f|Y) =h(f|X) and dimHY= dimHX.

The first identity in (3) was first established by Pesin and Pitskel in [10] for the full shift on two symbols. In a related direction, Shereshevsky [11] proved that for a genericC² surface diffeomorphism with a locally maximal hyperbolic setX and an equilibrium measureμof a H¨older continuousC⁰-generic function, the set of points for which the pointwise dimension does not exist has positive Hausdorff dimension, that is,

dim_H

x∈X: lim inf

r→0

logμ(B(x, r))

logr <lim sup

r→0

logμ(B(x, r)) logr

>0.

It was shown in [4] that the identities in (2) also hold for topologically mixing topological Markov chains and for repellers of C^1+ε maps. We refer the reader to [1, Chapter 8] for a detailed discussion of some of these results.

For topological Markov chains, the first identity in (2) was extended by Fan, Feng and Wu in [7] to arbitrary continuous functions. For repellers ofC^1+εconfor- mal maps, the second identity was extended by Feng, Lau and Wu in [8] to arbitrary continuous functions. For further related work, we refer the reader to [3] for anal- ogous results for hyperbolic flows, to [5] and [12] for the study of the entropy of irregular sets of continuous functions for maps with the specification property and to [9] for an extension of some of these results to the general case of the setsX_I.

2. Main result

Letσbe the shift map on Σ={1, ..., k}^Z, wherek≥2 is an integer. We equip Σ with the distanceddefined by

d ω, ω

= 2^{−inf{|n|:ωn=ωn}}, ω= (ωi)i∈Z, ω= ω_i

i∈Z. Given ak×k matrixA=(aij) with entries in {0,1}, let

ΣA=

(...ω₋1ω0ω1...)∈Σ :aω_nω_n+1= 1 forn∈Z .

The restriction of the shift mapσ|Σ_A: Σ_A→Σ_Ais called the (two-sided) topological Markov chain with transition matrixA. We recall thatσ|Σ_Ais topologically mixing if and only if some power ofA has only positive entries.

Given continuous functionsϕ, ψ: ΣA→R, we consider the level sets B_ϕ⁻(α) =

ω∈Σ_A: lim

n→∞S_ϕ⁻(ω, n) =α

and

B_ψ⁺(α) =

ω∈Σ_A: lim

n→∞S_ψ⁺(ω, n) =α

, where

(4) S_ϕ⁻(ω, n) = 1 2n−1

n−1 i=−n+1

ϕ

σ⁻ⁱ(ω)

, S⁺_ψ(ω, n) = 1 2n−1

n−1

i=−n+1

ψ σⁱ(ω)

.

Following arguments in [6], one can show that the sets L⁻ϕ=

α∈R:B_ϕ⁻(α)=∅

and L⁺_ψ=

α∈R:B_ψ⁺(α)=∅

are nonempty closed intervals. For each ω∈Σ_A, we denote byA⁻_ϕ(ω) and A⁺_ψ(ω), respectively, the sets of accumulation points of the sequences

n−→S_ϕ⁻(ω, n) and n−→S_ψ⁺(ω, n).

One can easily verify that A⁻_ϕ(ω) =

lim inf

n→∞ S_ϕ⁻(ω, n),lim sup

n→∞ S_ϕ⁻(ω, n)

and

A⁺_ψ(ω) =

lim inf

n→∞ S_ψ⁺(ω, n),lim sup

n→∞ S_ψ⁺(ω, n)

. The following is our main result.

Theorem 2.1. Let σ|ΣA be a topologically mixing topological Markov chain and let ϕ, ψ: ΣA→R be continuous functions. Given closed intervals I⊂L⁻ϕ and J⊂L⁺_ψ that are not singletons, if the set

Σ^ϕ,ψ_I,J :=

ω∈ΣA:A⁻_ϕ(ω) =I andA⁺_ψ(ω) =J is nonempty, then it is residual.

Proof. We first introduce some notation. For eachn∈N, let S_n=

(ω_−n...ω₀...ω_n) :a_ωiωi+1= 1 for−n≤i < n and let Σ^∗=

n∈NSn. Givenω=(...ω₋1ω0ω1...)∈ΣA, we writeω⁺=(ω0ω1...).More- over, givenω=(...ω₋₁ω₀ω₁...)∈Σ_Aandm∈Nor givenω=(ω_−n...ω_n)∈S_n andm∈N withm≤n, we writeω|m=ω_−m...ω_m.For eachω∈S_n, we write|ω|=2n+1 and we define the cylinder set

[ω] ={ρ∈Σ_A:ρ|n=ω}.

Givenω=(ω₋n...ωn)∈Sn andω=(ω_{− m}...ω_m)∈Sm, we write ωω=

ω₋n...ωnω_{− m}...ω_m .

Since all entries ofA^τ+1are positive, for any two admissible stringsω, ω∈Σ^∗ there existsρ=ρ(ω, ω)∈Sτ such thatωρω∈Σ^∗.We say thatρ(ω, ω) is abridgebetween ω and ω, and we write ωρω=ω ω (although we emphasize that ρ need not be unique). Moreover, given setsW, W₁, ..., W_n⊂Σ^∗ and a stringω∈Σ^∗, we write

W1 ... Wn={ω1 ω2 ... ωn:ωi∈Wi,1≤i≤n} and

ω W={ω η:η∈W},

where each symbol runs over all admissible bridges. Finally, we write Wⁿ=W1 ... Wn

whenW1=...=Wn=W.

We proceed with the proof of the theorem. It consists of constructing a dense G_δ set E⊂Σ_A such thatE⊂Σ^ϕ,ψ_I,J. Given α∈R,ε>0 andn∈N, let

L(α, n, ε) =

ω|n−1:ω∈Σ_A andS_ϕ⁻(ω, n)−α< ε and

R(α, n, ε) =

ω|n−1:ω∈ΣAandS_ψ⁺(ω, n)−α< ε , withS_ϕ⁻(ω, n) andS_ψ⁺(ω, n) as in (4). Clearly, for eachε>0, we have

L(α, n, ε)=∅ and R(β, n, ε)=∅ forα∈L⁻ϕ, β∈L⁺ψ and all sufficiently large n.

Now takek∈Nand choose αk,1, ..., αk,q_k∈I andβk,1, ..., βk,q_k∈J such that I⊂

q_k

i=1

B

αk,i,1 k

, J⊂

q_k

i=1

B

βk,i,1 k

and

|α_k,i+1−α_k,i|<1

k, |α_k,qk−α_k+1,1|<1 k,

|βk,i+1−βk,i|<1

k, |βk,q_k−βk+1,1|<1 k (5)

fori=0, ..., qk−1. Moreover, we consider a sequence of positive real numbers ε_1,1> ε1,2> ... > ε1,q1> ε2,1> ε2,2> ... > ε2,q2> ...

decreasing to zero and a sequence

n1,1< n1,2< ... < n1,q1< n2,1< n2,2< ... < n2,q2< ...

of positive integers such that

Lk,i:= (αk,i, nk,i, εk,i)=∅ and Rk,i:=R(βk,i, nk,i, εk,i)=∅ fork∈Nand 1≤i≤qk.

Let Ω0=Σ^∗. For each ω∈Ω0, we choose positive integers Nk,i=Nk,i(ω) for k∈Nand i=1, ..., q_k such that:

(i) N_1,i≥2^m1,i+1+τ for 2≤i≤q₁−1, Nk,i≥2^mk,i+1+τ fork≥2, 1≤i≤qk−1, Nk,q_k≥2^mk+1,1+τ fork≥1;

(ii) N_k,i+1≥2^{|ω|+τ+N1,1(m1,1+τ)+N1,2(m1,2+τ)+...+Nk,i(mk,i+τ)},

N_k+1,1≥2^{|ω|+τ+N1,1(m1,1+τ)+N1,2(m1,2+τ)+...+Nk,qk(mk,qk+τ)} fork≥1, 1≤i<qk.

Here mi,k=2ni,k−1 and τ is some fixed integer such that A^τ+1 has only positive entries. We define recursively sets Ω_k,i⊂Σ^∗ fork∈Nandi=1, ..., q_k by

Ω1,1=

ω∈Ω0

L^N_1,1^1,1(ω) ω R^N_1,1^1,1(ω),

Ω1,2=

η∈Ω1,1

L^N_1,2^1,2(ω) η R^N_1,2^1,2(ω),

...

Ω_1,q1=

η∈Ω_1,q1−1

L^N_1,q^1,q1(ω)

1 η R^N_1,q^1,q1(ω)

1 ,

Ω2,1=

η∈Ω1,q1

L^N_2,1^2,1(ω) η R^N_2,1^2,1(ω),

and so on. Finally, let

Ek,i=

ω∈Ωk,i

[ω]

and

E=

∞ k=1

q_k

i=1

E_k,i.

We note that E is a Gδ set since each cylinder set [ω] is open. Moreover, since Ω0=Σ^∗, each setEk,i is dense and so, it follows from Baire’s theorem thatE is also dense.

Lemma 2.2. E⊂Σ^ϕ,ψ_I,J.

Proof. In order to prove that E⊂Σ^ϕ,ψ_I,J, we must show that A⁻_ϕ(ω)=I and A⁺_ψ(ω)=J for eachω∈E. We only show that

(6) A⁻_ϕ(ω) =I,

since the identityA⁺_ψ(ω)=J can be proved in a similar manner.

We first show that

(7) I⊂A⁻_ϕ(ω).

Given α∈I⊂qk

i=1B(αk,i,1/k), take ik∈{1, ..., qk}such that α∈B(αk,i_k,1/k). For simplicity of the exposition we assume thati_k∈{1, q_k}. We recall that there exists ω⁰∈Ω₀ such that

(8) ω∈... L^N_1,1^1,1 ω⁰ R^N_1,1^1,1 ..., whereN_k,i=N_k,i(ω⁰). Let

(9) sk,i_k=ω⁰+τ+

q1

j=1

N1,j(m1,j+τ)+...+

i_k

j=1

Nk,j(mk,j+τ).

We will prove that

(10) S_ϕ⁻(ω, sk,i_k)−αk,i_k→0 whenk→∞. It then follows from (10) that

S_ϕ⁻(ω, sk,i_k)−α≤S_ϕ⁻(ω, sk,i_k)−αk,i_k+|αk,i_k−α|

<S_ϕ⁻(ω, sk,i_k)−αk,i_k+1 k→0 whenk→∞. Therefore,α∈A⁻_ϕ(ω) and (7) holds. Write (11) s_k,ik= ˜sk,ik+tk,ik,

wheretk,i_k=Nk,i_k(mk,i_k+τ).Since|αk,i_k|≤ϕ, whereϕ=maxω∈ΣA|ϕ(ω)|, writ- ingmk,i_k+τ=r we obtain

s_k,ik−1

i=0

ϕ

σ⁻ⁱ(ω)

−sk,i_kαk,i_k

≤

˜ s_k,ik−1

i=0

ϕ

σ⁻ⁱ(ω)

−˜sk,i_kαk,i_k

+

s_k,ik−1

i=˜s_k,ik

ϕ

σ⁻ⁱ(ω)

−tk,i_kαk,i_k

≤2˜s_k,ikϕ+

t_k,ik−1

i=0

ϕ σ⁻ⁱ

σ^−s˜k,ik(ω)

−t_k,ikα_k,ik

≤2˜sk,i_kϕ+2τ Nk,i_kϕ +

N_k,ik−1

q=0

m_k,ik−1

j=0

ϕ σ^−j

σ^{−(˜sk,ik+qr)}(ω)

−mk,i_kαk,i_k

= 2˜s_k,ikϕ+2τ N_k,ikϕ +

N_k,ik−1

q=0

m_k,ik−1

j=−m_k,ik+1

ϕ σ^−j

σ^{−(˜sk,ik+nk,ik−1+qr)}(ω)

−m_k,ikα_k,ik . (12)

We introduce the numberVn(ϕ)=2n−1

j=0Varj(ϕ), where Var_k(ϕ) = supϕ(ω)−ϕ

ω:ω, ω∈Σ_A, ω|k=ω|k

.

By (8) and the definition ofLk,i_k, there existω⁰, ..., ω^Nk,ik−1∈ΣA such that (13) σ^{−(˜sk,ik+nk,ik−1+qr)}(ω)|n_k,ik−1=ω^q|n_k,ik−1

and

(14) S_ϕ⁻

ω^q, nk,i_k

−αk,i_k< εk,i_k

forq=0, ..., Nk,i_k−1.

It follows from (13) and (14) that

n_k,ik−1

j=−n_k,ik+1

ϕ σ^−j

σ^{−(˜sk,ik+nk,ik−1+qr)}(ω)

−mk,i_kαk,i_k

≤

n_k,ik−1

j=−n_k,ik+1

ϕ σ^−j

σ^{−(˜sk,ik+nk,ik−1+qr)}(ω)

−

n_k,ik−1

j=−n_k,ik+1

ϕ σ^−j

ω^q

+

n_k,ik−1

j=−n_k,ik+1

ϕ σ^−j

ω^q

−m_k,ikα_k,ik

≤V_nk,ik(ϕ)+m_k,ikε_k,ik

forq=0, ..., N_k,ik−1. Together with (12) this implies that

s_k,ik−1

i=0

ϕ

σ⁻ⁱ(ω)

−sk,i_kαk,i_k

≤2˜s_k,ikϕ+N_k,ik

V_nk,ik(ϕ)+m_k,ikε_k,ik

+2τ N_k,ikϕ

= 2˜sk,i_kϕ+Nk,i_kVn_k,ik(ϕ)+Nk,i_k

mk,i_kεk,i_k+2τϕ . Using (9), (11) and condition (ii), we obtain

sk,ik

˜

sk,i_k−1≥2^s˜k,ik

˜ sk,i_k

(mk,i_k+τ) and thus,

sk,i_k

˜

sk,i_k →+∞ whenk→∞.

Moreover, it follows from the uniform continuity of ϕon the compact set ΣA that Varn(ϕ)→0 whenn→∞. Hence,Vn(ϕ)/n→0 whenn→∞and

Nk,i_kVn_k,ik(ϕ) sk,i_k

≤Vn_k,ik(ϕ) mk,i_k

→0 whenk→∞. By the definition ofs_k,ik, we haves_k,ik>t_k,ik and

Nk,i_k

sk,ik

<Nk,i_k

tk,ik

= 1

mk,ik+τ. Therefore,

S_ϕ⁻(ω, sk,i_k)−αk,i_k<2˜sk,i_kϕ

s_k,ik +Vn_k,ik(ϕ)

m_k,ik +mk,i_kεk,i_k+2τϕ n_k,ik →0 whenk→∞, which completes the proof of (10).

Now we show that

(15) A⁻_ϕ(ω)⊂I.

For each positive integer n>|ω⁰|+τ there exist k∈N, i_k∈{1,2, ..., q_k} and 0≤p<

Nk,ik+1−1 such that

(16) s_k,ik+p(mk,ik+1+τ)< n≤sk,ik+(p+1)(mk,ik+1+τ).

We claim that

(17) S_ϕ⁻(ω, n)−α_k,ik→0 whenn→∞.

Again for simplicity of the exposition, we assume thatik=qk. If (17) holds, then it follows from (5) that

dist

S_ϕ⁻(ω, n), I

≤S_ϕ⁻(ω, n)−αk,i_k+dist(αk,i_k, I)→0

whenn→∞(notice thatk→∞ whenn→∞). SinceI is closed, we conclude that A⁻_ϕ(ω)⊂I and (15) holds.

Now we establish (17). Writingmk,i_k+1+τ=r, we obtain

n−1 i=0

ϕ

σ⁻ⁱ(ω)

−nαk,i_k

≤

s_k,ik−1

i=0

ϕ

σ⁻ⁱ(ω)

−sk,i_kαk,i_k

+

s_k,ik+pr−1

i=s_k,ik

ϕ

σ⁻ⁱ(ω)

−prαk,i_k

+

n−1 i=s_k,ik+pr

ϕ

σ⁻ⁱ(ω)

−(n−s_k,ik−pr)α_k,ik .

As in (13) and (14), one can chooseω⁰, ..., ω^p−1∈ΣAsuch that (18) σ^{−(sk,ik+nk,ik+1−1+qr)}(ω)|n_k,ik+1−1=ω^q|n_k,ik+1−1

and

(19) S⁻_ϕ

ω^q, nk,i_k+1

−αk,i_k+1< εk,i_k+1

forq=0, ..., p−1. It follows from (5), (18) and (19) that

m_k,ik+1−1

j=0

ϕ σ^−j

σ^{−(sk,ik+qr)}(ω)

−m_k,ik+1α_k,ik

≤

m_k,ik+1−1

j=0

ϕ σ^−j

σ^{−(sk,ik+qr)}(ω)

−m_k,ik+1α_k,ik+1 +|m_k,ik+1αk,ik+1−mk,ik+1αk,ik|

≤

n_k,ik+1−1

j=−n_k,ik+1+1

ϕ σ^−j

σ^{−(sk,ik+nk,ik+1−1+qr)}(ω)

−

n_k,ik+1−1

j=−n_k,ik+1

ϕ σ^−j

ω^q +

n_k,ik+1−1

j=−n_k,ik+1+1

ϕ σ^−j

ω^q

−mk,i_k+1αk,i_k+1

+mk,i_k+1

k

≤V_nk,ik+1(ϕ)+m_k,ik+1ε_k,ik+1+m_k,ik+1 k forq=0, ..., p−1. Therefore,

s_k,ik+pr−1

i=s_k,ik

ϕ

σ⁻ⁱ(ω)

−prαk,i_k

≤

p−1

q=0

n_k,ik+1−1

j=−n_k,ik+1+1

ϕ σ^−j

σ^{−(sk,ik+qr)}(ω)

−mk,i_k+1αk,i_k

+2pτϕ

≤p

Vn_k,ik+1(ϕ)+mk,i_k+1εk,i_k+1+mk,i_k+1

k

+2pτϕ. (20)

Moreover, by (16), we have

n−1 i=s_k,ik+pr

ϕ

σ⁻ⁱ(ω)

−(n−s_k,ik−pr)α_k,ik

≤2(n−s_k,ik−pr)ϕ ≤2rϕ= 2(m_k,ik+1+τ)ϕ. (21)

By (20) and (21), we obtain

(22)

S_ϕ⁻(ω, n)−α_k,ik≤S_ϕ⁻(ω, s_k,ik)−α_k,iksk,i_k

n +2(m_k,ik+1+τ)ϕ

n +pVm_k,ik+1(ϕ) n +pm_k,ik+1

kn +p(m_k,ik+1ε_k,ik+1+2τϕ)

n .

As in the proof of (10), one can show that the first term in the right-hand side of (22) tends to zero when n→∞ (notice that sk,i_k≤n). Moreover, using (16) and condition (i), we obtain

2(m_k,ik+1+τ)ϕ

n ≤2(m_k,ik+1+τ)ϕ

sk,i_k ≤2(m_k,ik+1+τ)ϕ Nk,i_k →0

whenn→∞. On the other hand, it follows from (16) that pm_k,ik+1

kn ≤2

k→0 whenn→∞, pVn_k,ik+1(ϕ)

n ≤Vn_k,ik+1(ϕ) mk,i_k+1

→0 whenn→∞, and

p(m_k,ik+1ε_k,ik+1+2τϕ)

n ≤m_k,ik+1ε_k,ik+1+2τϕ

mk,i_k+1 →0 whenn→∞ (since k→∞ when n→∞). This shows that the right-hand side of (22) tends to zero whenn→∞, which establishes (17).

Property (6) follows now readily from (7) and (15) and the proof of Lemma2.2 is complete.

This shows that the set E has the desired properties and the proof of the theorem is complete.

The following is a simple consequence of Theorem 2.1.

Theorem 2.3. Let σ|ΣA be a topologically mixing topological Markov chain and letψ: ΣA→Rbe a continuous function. Given a closed interval J⊂L⁺_ψ that is not a singleton,if the set

Σ^ϕ_I :=

ω∈Σ_A:A⁺_ψ(ω) =I is nonempty, then it is residual.

Proof. It suffices to observe that Σ^ϕ,ψ_I,J⊂Σ^ψ_J and apply Theorem2.1.

As an application of Theorem2.3, we obtain the following result.

Theorem 2.4. Let σ|ΣA be a topologically mixing topological Markov chain and letϕ: ΣA→Rbe a continuous function. If the irregular set

Σ^ϕ:=

ω∈Σ_A: lim inf

n→∞ S_ϕ⁺(ω, n)<lim sup

n→∞ S⁺_ϕ(ω, n) is nonempty, then it is residual.

Proof. If the set Σ^ϕis nonempty, then there exists a closed intervalI⊂Lϕwith Σ^ϕ_I=∅that is not a singleton. Otherwise, if Σ^ϕ_I=∅for any closed interval, then Σ^ϕ would be empty (since Σ^ϕ_I=∅whenI is not a closed interval), and if for any closed interval I such that Σ^ϕ_I=∅this last set was a singleton, then again Σ^ϕ would be empty. Since Σ^ϕ_I⊂Σ^ϕ, the desired result follows readily from Theorem2.3.

3. Results for hyperbolic sets

In this section we obtain corresponding results for the Birkhoff averages of a continuous function on a locally maximal hyperbolic set.

Letf:M→M be aC¹diffeomorphism on a smooth manifoldM and let Λ⊂M be a compactf-invariant set. We say thatf is ahyperbolic set forf if there exist τ∈(0,1),c>0 and a decomposition

TxM=E^s(x)⊕E^u(x) for eachx∈Λ, such that

dxf E^s(x) =E^s f(x)

, dxf E^u(x) =E^u f(x)

, dxfⁿv≤cτⁿv wheneverv∈E^s(x), and dxf⁻ⁿv≤cτⁿv wheneverv∈E^u(x)

for everyx∈Λ andn∈N. We say that Λ islocally maximal if there exists an open setU⊃Λ such that

Λ =

n∈Z

fⁿ(U).

Given continuous functionsϕ, ψ: Λ→Rand setsI, J⊂R, let Λ^ϕ,ψ_I,J =

x∈X:A⁻_ϕ(x) =I, A⁺_ψ(x) =J ,

where A⁻_ϕ(x) and A⁺_ψ(x) are, respectively, the sets of accumulation points of the sequences

n−→S_ϕ⁻(x, n) and n−→S_ψ⁺(x, n), withS_ϕ⁻(x, n) andS⁺_ψ(x, n) as in (4). Moreover, let

R⁻ϕ=

α∈R:B_ϕ⁻(α)=∅

and R⁺_ψ=

α∈R:B⁺_ψ(α)=∅ , where

B⁻_ϕ(α) =

x∈Λ : lim

n→∞S_ϕ⁻(x, n) =α

and

B_ψ⁺(α) =

x∈Λ : lim

n→∞S_ψ⁺(x, n) =α

.

The following is a version of Theorem2.1for the Birkhoff averages on a locally maximal hyperbolic set.

Theorem 3.1. Let Λ be a locally maximal hyperbolic set for a topologically mixing C¹ diffeomorphism f. Given closed intervals I⊂R⁻ϕ and J⊂R⁺_ψ that are not singletons,if the set Λ^ϕ,ψ_I,J is nonempty,then it is residual.

Proof. Since Λ is a locally maximal hyperbolic set, there existsδ >0 such that the map

[·,·] :

(x, y)∈Λ×Λ :d(x, y)≤δ

→Λ

is well defined. We recall that a closed setR⊂Λ is called a rectangle if:

1. diamR<δ andR=intR, with the interior taken with respect to the induced topology on Λ;

2. [x, y]∈Rwheneverx, y∈R.

Moreover, a collection of rectanglesR1, ..., Rk⊂Λ is called a Markov partition of Λ (with respect tof) if:

1. intR_i∩intR_j=∅wheneveri=j;

2. ifx∈intR_i andf(x)∈intR_j, then f

V_i^u(x)

⊃V_j^u f(x)

and f⁻¹ V_j^s

f(x)

⊃V_i^s(x), where

V_i^s(x) =

y∈B(x, ε) :d

fⁿ(x), fⁿ(y)

< εforn≥0

∩R_i, V_i^u(x) =

y∈B(x, ε) :d

fⁿ(x), fⁿ(y)

< εforn≤0

∩Ri,

and whereB(x, ε) is the ball of radiusε centered atx(for some sufficiently small ε>0).

Any locally maximal hyperbolic set has Markov partitions of arbitrarily small di- ameter (see for example [2]). Let A=(aij) be a k×k matrix with entries aij=1 if intf(Ri)∩intRj=∅ and aij=0 otherwise. Writing X=ΣA, we obtain a coding mapπ:X→Λ for the set Λ letting

π(ω) =

n∈Z

f⁻ⁿR_ωn, ω= (...ω₋₁ω₀ω₁...).

One can verify that π is continuous, onto and that π◦σ=f◦π in X. This last identity implies thatL⁻ϕ◦π=R⁻ϕ andL⁺ψ◦π=R⁺ψ.

Now let

B=

n∈Z

fⁿ _k

i=1

∂R_i

=

n∈Z

k i=1

fⁿ(∂R_i),

where∂Ri is the boundary ofRi, be the set of points in Λ for which the coding is not unique. We consider the sets

S=X\π⁻¹B and Λ^∗= Λ\B.

Clearly, the map π: S→Λ^∗ is bijective. Moreover, ∂Ri is closed and has empty interior. Since f is a diffeomorphism, each set fⁿ(∂Ri) is closed and has empty interior. Hence, B is an Fσ set and by Baire’s theorem it has empty interior.

Moreover, since πis continuous,S=π⁻¹Λ^∗ is aG_δ set.

Now we show that S is dense. We proceed by contradiction. If S was not dense, then it would exist a cylinder set [i₋m...im]⊂X\S=π⁻¹B. Therefore,

R:=

m

n=−m

f⁻ⁿ⁺¹R_in=π

[i_−m...i_m]

⊂π π⁻¹B

=B,

sinceπis onto. But by the properties of a Markov partition, this finite intersection has nonempty interior (indeed, R=intR), which contradicts the fact that B has empty interior. Hence,S is dense.

We note that Φ=ϕ◦πand Ψ=ψ◦πare continuous functions onX. Let I⊂ R⁻ϕ=L⁻_Φ and J⊂ R⁺_ψ=L⁺_Ψ

be closed intervals. It follows from Theorem 2.1 that there exists a dense Gδ set E⊂X_I,J^Φ,Ψ. To complete the proof, it suffices to show that the setF=π(E∩S)⊂Λ^∗ satisfies the following properties:

1. F⊂Λ^ϕ,ψ_I,J; 2. F is dense in Λ;

3. F is aG_δ set.

It follows from the identityπ◦σ=f◦πthat F⊂π(E)⊂π

X_I,J^Φ,Ψ

= Λ^ϕ,ψ_I,J.

Moreover, E∩S is a dense G_δ set since both E and S are dense G_δ sets. In particular,

Λ =π(X) =π(E∩S)⊂π(E∩S) =F