RECURRENCE SPECTRUM IN SMOOTH DYNAMICAL SYSTEMS.

BENOÎT SAUSSOL AND JUN WU

Abstract. We prove that for conformal expanding maps the return time does have constant multifractal spectrum. This is the counterpart of the result by Feng and Wu in the symbolic setting.

1. Introduction

Let f : ℳ → ℳ be a map on a smooth compact manifold ℳ with metric d. Let Λ ⊂ ℳ be a compact f-invariant set. For x ∈ ℳ we can define the return times in r-neighborhoods by

τ_{r}(x) = inf{n > 0: d(f^{n}x, x) < r}.

We put τ_{r}(x) = ∞ if the orbit of x does not come back close to x.

When ℳ is an interval, f is a piecewise monotonic transformation with some regularity (piecewise C^{1+α} suffices) and μ is an f-invariant ergodic measure with nonzero entropy, Saussol, Troubetzkoy and Vaienti [11], building on results by Ornstein and Weiss [10] and Hofbauer and Raith [5, 6], proved that

(1)  lim_{r→0} \frac{log τ_{r}(x)}{− log r} = dim_{H} μ for μ-almost every x

where dim_{H} μ stands for the Hausdorff dimension of the measure μ. Barreira and Saussol established, with completely different techniques, Equation (1) for equilibrium states preserved by repellers [2] and Axiom A diffeomorphisms [1] in any dimension.

From now on we assume that Λ ⊂ ℳ is a conformal repeller of a C^{1+α} map f: ℳ → ℳ. That is to say that Λ is a compact forward invariant set and that f is C^{1+α} on ℳ, expanding and conformal on Λ. Moreover we suppose that there exists an open set V ⊂ ℳ such that Λ = ⋂_{n≥0} f^{−n}V.

From the multifractal analysis of conformal expanding maps it is well known that for any real s ∈ [0, dim_{H} Λ] there exists an ergodic equilibrium state μ_{s} (in fact there are many such measures for non extremal values of s). Therefore Equation (1) together with basic properties of dimension of measures imply that

(2) dim_{H} \{x ∈ Λ: lim_{r→0} \frac{log τ_{r}(x)}{− log r} exists and equals s\} ≥ dim_{H} μ_{s} = s.

The reader familiar with multifractal analysis will believe that in fact the dimension of this set is larger than this, and at this point may expect a nice non-trivial (strictly) concave analytic curve. In fact, this is really not the case, as our first theorem says

Date: March 28, 2003.

1991 Mathematics Subject Classification. Primary: 37B20, 37C45, 37D20.

Key words and phrases. Poincaré recurrence, multifractal analysis.

Theorem 1. Let Λ ⊂ ℳ be a conformal repeller of the C^{1+α} map f: ℳ → ℳ. For any α ≤ β in [0, ∞] we have

dim_{H} \{x ∈ Λ: \underline{lim}_{r→0} \frac{log τ_{r}(x)}{− log r} = α and \overline{lim}_{r→0} \frac{log τ_{r}(x)}{− log r} = β\} = dim_{H} Λ.

Theorem 1 implies in particular that for all values of s ∈ [0, ∞] the dimension in the left hand side of Equation (2) equals dim_{H} Λ.

The corresponding result of Theorem 1 in the symbolic case was established by Feng and Wu [3]. Although our method relies on some essential ideas present in the above mentioned paper, we emphasize that our proof is not a direct translation of the symbolic result. ^{1}

In Section 2 we prove the theorem in the situation where the system (f, Λ) is conjugated to a subshift of finite type. It happens that our problem can be reduced to this case and this is proven in Section 3.

Contents

1. Introduction 1 2. System conjugated to a subshift of finite type 2 2.1. Producing points with given symbolic recurrence sequence 3 2.2. Building the source of large dimension 3 2.3. The image of the source has good symbolic recurrence 5 2.4. The image has large dimension 6 2.5. The image has good recurrence rates 7 3. Reduction to a system conjugated to a subshift 7 3.1. Pressure for systems with small hole 8 3.2. Removing the boundary of the Markov partition 10 4. Proofs of the technical lemmas 10 Acknowledgments 12 References 12

2. System conjugated to a subshift of finite type

In this section we give a proof of Theorem 1 in a special case.

Theorem 2. Let Λ ⊂ ℳ be such that (f, Λ) is conjugated to a subshift of finite type. Then the result of Theorem 1 holds.

The proof of this result is decomposed in several parts. First we give a construction similar to the one by Feng and Wu of points with prescribed recurrence in a symbolic space [3].

1After the completion of this work we learn that L. Olsen has obtained — independently and with different techniques — a similar result for conformal Iterated Function Systems satisfying some strong separation condition, which is equivalent to a conjugacy to a subshift of finite type, the case considered in Theorem 2.

2.1. Producing points with given symbolic recurrence sequence. For a given sequence of integer (ℓ_{n}) such that

(a) ∃n_{0}, ℓ_{n+1} ≥ ℓ_{n} + 2n for any n ≥ n_{0} and

(b) lim_{n→∞} ℓ_{n}/n^{2} = ∞, we define a function g nearly as in [3] such that points in the image of g have (ℓ_{n}) as recurrence sequence. Let 𝒜 ⊂ ℬ be two alphabets (i.e. finite or countable sets) with 𝒜 ≠ ℬ and ♯ℬ ≥ 3. Choose a marker m ∈ ℬ \ 𝒜.

Definition 3. Let c ≠ c̄ ∈ ℬ\{m}. Define g: 𝒜^{ℕ} → ℬ^{ℕ} by g(w) = lim w^{(k)}, where the sequence w^{(k)} is constructed recursively as follows.

We put w^{(0)} = w^{(1)} = ⋯ = w^{(n_{0}−1)} = mw_{1}w_{2} · · ·. For k ≥ n_{0} we construct w^{(k)} from w^{(k−1)} by inserting the block w_{1}^{(k−1)} ⋯ w_{k}^{(k−1)}y_{k} at the position ℓ_{k} + 1, i.e.

w^{(k)} = w_{1}^{(k−1)}w_{2}^{(k−1)} ⋯ w_{ℓ_{k}}^{(k−1)}w_{1}^{(k−1)} ⋯ w_{k}^{(k−1)}y_{k}w_{ℓ_{k}+1}^{(k−1)}w_{ℓ_{k}+2}^{(k−1)} ⋯,

where y_{k} = c if w_{k+1}^{(k−1)} ≠ c or y_{k} = c̄ if w_{k+1}^{(k−1)} = c, so that y_{k} ≠ m, w_{k+1}^{(k−1)}. Note that all the w^{(k)} share the same ℓ_{n+1} first letters whenever k ≥ n, hence the limit g(w) is well defined. □

Let us define the k-repetition time of w by

R_{k}(w) = inf{n > 0: w_{n+1}w_{n+2} ⋯ w_{n+k} = w_{1}w_{2} ⋯ w_{k}}.

The main interest of the function g resides in the following

Lemma 4 ([3]). If w ∈ 𝒜^{ℕ} and k ≥ n_{0} then R_{k}(g(w)) = ℓ_{k}.

Proof of Lemma 4. We do it by induction. Let w^{∗} = g(w). Since there is no m in w it is obvious that R_{n_{0}}(w^{∗}) = ℓ_{n_{0}}. Assume that for some k ≥ n_{0} we have proven that R_{k}(w^{∗}) = ℓ_{k}. By construction this implies that ℓ_{k} < R_{k+1}(w^{∗}) ≤ ℓ_{k} + k or R_{k+1}(w^{∗}) = ℓ_{k+1} (note that there can be no m between the positions ℓ_{k} + k + 1 and ℓ_{k+1} in w^{∗}). Suppose that we are in the first case and let p = R_{k+1}(w^{∗}). The number of m in the block w_{p+1}^{∗} ⋯ w_{p+k}^{∗} = w_{1}^{∗} ⋯ w_{k}^{∗} being equal to the number of m inside the block w_{ℓ_{k}+1}^{∗} ⋯ w_{ℓ_{k}+k}^{∗} we get a contradiction if p > ℓ_{k} (because w_{ℓ_{k}+1}^{∗} = m is missed and there is no m in w_{ℓ_{k}+k+1}^{∗} ⋯ w_{p+k}^{∗}). □

Although one can imagine similar constructions for subshifts of finite type, we decided to work with fullshifts to make the core argument more transparent. However, we then have to work a bit to apply it to our systems, which are presumably only coded by subshifts of finite type. This is the aim of next section.

2.2. Building the source of large dimension. Suppose that dim_{H} Λ > 0 otherwise there is nothing to prove in Theorem 2. Let (Σ, T) be a subshift of finite type conjugated to (f, Λ) and denote by π: Σ → Λ the conjugating homeomorphism. For our purpose there is no loss of generality to suppose that (Σ, T) is topologically mixing. Thus at least one letter, say a, has two successors b and c. Let then B and C be the shortest paths (and smallest in the lexicographic order) starting in b and c, respectively, and ending in a. That is aB and aC are the smallest words of Σ with prefix ab and ac, respectively, and suffix a. It is clear that there is a one-to-one correspondence between the set of words and the set of cylinders. Without confusion, we also use a word to denote it’s corresponding cylinder. Let A = aB and denote by t(ω) the return time of ω ∈ A into A,

t(ω) = inf{t > 0: T^{t}ω ∈ A}.

Note that t is unbounded on A since there is a disjoint union contained in A. More precisely,

A ⊃ aBB ∪ aBCB ∪ aBCCB ∪ · · · .

To simplify the exposition we will consider the new alphabet consisting in |A|-cylinders, where |A| denotes the length of the cylinder A, together with its associated transition matrix. Let then Z_{n} be the partition of Σ by ncylinders (the new ones), and denote by ℱ_{n} the finite σ-algebra generated by Z_{n}.

Let μ be the equilibrium state of the potential φ = − dim_{H}(Λ) log |Df|∘π. It is well known that μ is the measure of maximal dimension and the Bowen pressure formula gives P (φ|Σ) = 0. In addition the measure μ is ψ-mixing (See e.g. [7] for properties of equilibrium states), which means that

ψ(n) =^{def} sup_{m} sup_{U∈ℱ_{m},V ∈ℬ} |\frac{μ(U ∩ T^{−n−m}V)}{μ(U)μ(V)} − 1| ⟶_{n→∞} 0.

Hence there exists m_{0} such that

μ(A ∩ T^{−m_{0}}(Σ \ A) ∩ T^{−2m_{0}}(Σ \ A) ∩ ⋯ ∩ T^{−mm_{0}}(Σ \ A)) ≤ δ^{m},

where δ =^{def} μ(Σ \ A)(1 + ψ(m_{0})) < 1. Set H_{n} = {ω ∈ A: t(ω) ≥ n}. The preceding estimate yields

μ(H_{n}) ≤ δ^{n/m_{0}}.

Moreover we have H_{n} = A\{x ∈ A: t(x) ≤ n−1} ∈ ℱ_{n}. From Proposition 7 (see Section 3 below), setting Σ_{n} = (Σ \ H_{n})^{∞}, we get that

(3)  lim_{n→∞} P (φ|Σ_{n}) = P (φ|Σ) = 0.

Let ν_{n} be an ergodic equilibrium state of φ on Σ_{n}. If n is sufficiently large then ν_{n}(A) > 0 (otherwise ν_{n} would be supported on the forward invariant set (Σ \ A)^{∞}, on which the pressure of φ is strictly less than P (φ|Σ)).

Let Σ̂ ⊂ A denotes the set of points returning infinitely many times in A. Let us define the induced system (Σ̂, T̂, ν̂_{n}) by

T̂(x) = T^{t(x)}(x), ν̂_{n} = \frac{1}{ν_{n}(A)}ν_{n}|_{A}.

Since T̂ is the induced transformation on a cylinder of a subshift of finite type, (Σ̂, T̂) is a full shift with the countable alphabet

Ẑ = ⋃_{m=1}^{∞}{Z ∩ Σ̂: Z ∈ Z_{m+1}, Z ⊂ A, t|_{Z} = m}.

Since the return times in A are bounded (by n − 1) on Σ \ H_{n}, one has

Σ̂_{n} =^{def} A ∩ Σ_{n} ⊂ Σ̂.

Observe that (Σ̂_{n}, T̂) is still a fullshift over the finite alphabet

Ẑ^{n} = ⋃_{m=1}^{n−1}{Z ∩ Σ̂_{n}: Z ∈ Z_{m+1}, Z ⊂ A, t|_{Z} = m}.

Set ψ = log |Df| ∘ π, S_{k}ψ(ω) = ∑_{j=0}^{k−1} ψ(T^{j}ω) and Ŝ_{k}t(ω) = ∑_{j=0}^{n−1} t(T̂^{j}ω). Let

us define the source of large dimension by

G_{n} = \{ω ∈ Σ̂_{n}: (i) lim_{k→∞} \frac{1}{k} S_{k}ψ(ω) → ν_{n}(ψ), (ii) lim_{k→∞} \frac{1}{k} Ŝ_{k}t(ω) → ν̂_{n}(t)\} .

By Birkhoff’s Ergodic Theorem we have ν̂_{n}(G_{n}) = 1 which provides the following lower bound

(4)  dim_{H} πG_{n} ≥ dim_{H} πν̂_{n} = dim_{H} πν_{n}|_{A} = dim_{H} πν_{n}.

The last equality follows by ergodicity of the equilibrium state ν_{n}. Moreover,

dim_{H} πν_{n} = \frac{h_{ν_{n}}}{∫ ψdν_{n}} = dim_{H} Λ + \frac{P (φ|Σ_{n})}{∫ ψdν_{n}}.

Observe that this together with (3) and (4) imply that

(5)  lim_{n→∞} dim_{H} πG_{n} = dim_{H} Λ.

2.3. The image of the source has good symbolic recurrence. With the alphabets 𝒜 = Ẑ^{n} and ℬ = Ẑ we can use Definition 3 to produce a function g: Σ̂_{n} → Σ̂. For the marker m ∈ ℬ \ 𝒜 we choose a m ∈ Ẑ such that t|_{m} = n.

Given an integer k, let p be such that ℓ_{p} ≤ k < ℓ_{p+1} and set ϵ_{k} = (p + 2)^{2}/ℓ_{p}. The property (b) of the sequence (ℓ_{p}) implies that ϵ_{k} → 0 as k → ∞.

Observe that t is Ẑ measurable, hence for any ω ∈ G_{n} we have

(6)  |Ŝ_{k}t(gω) − Ŝ_{k}t(ω)| ≤ kϵ_{k}n,

since the symbolic sequence of g(ω) and ω differs only because of the block of length i inserted at position ℓ_{i}, which makes a difference of at most ∑_{i=n_{0}+1}^{p+1} i ≤ kϵ_{k} letters. Hence for any ω ∈ G_{n} we have

(7)  lim_{k→∞} \frac{1}{k} Ŝ_{k}t(gω) = ν̂_{n}(t).

Notice that a k-cylinder of (Σ̂, T̂) containing ω is indeed a Ŝ_{k}t(ω) cylinder of (Σ, T). Hence Lemma 4 gives that for ω ∈ G_{n} we have (the notation R̂_{k} stands for the k-repetition time computed with T̂)

R_{Ŝ_{k}t(gω)}(gω) = R̂_{k}(gω) = ℓ_{k}.

Passing to the limit using Equation (7) and the fact that the sequence (R_{h})_{h} is monotone gives us that the image gG_{n} has good symbolic recurrence. Namely, for any ω ∈ G_{n},

(8)  \underline{lim}_{h→∞} \binom{\overline{lim}}{h→∞} \frac{log R_{h}(gω)}{h} = \underline{lim}_{k→∞} \binom{\overline{lim}}{k→∞} \frac{1}{ν̂_{n}(t)} \frac{log ℓ_{k}}{k} .

2.4. The image has large dimension. The fact that the dimension of the image πgG_{n} is large will follow from (5) if we are able to prove that the inverse function of γ =^{def} π ∘ g ∘ π^{−1} has some smoothness.

Let ω ∈ G_{n}. Denotes the h-cylinder of (Σ, T, Z) containing g(ω) by Z(gω, h). If k is such that

Ŝ_{k}t(gω) ≤ h < Ŝ_{k+1}t(gω)

then

Z(gω, h) ⊂ Ẑ(gω, k),

where Ẑ(z, k) denotes the k-cylinder of (Σ̂, T̂, Ẑ) containing a point z. By construction of g we have

g^{−1}Ẑ(gω, k) ⊂ Ẑ(ω, k(1 − ϵ_{k})).

Because of (6) we have

Ẑ(ω, k(1 − ϵ_{k})) ⊂ Z(ω, h(1 − δ_{h}))

for some sequence δ_{h} → 0 as h → ∞ (independent of ω). Putting together this chain of inclusions gives us that

(9)  g^{−1}Z(gω, h) ⊂ Z(ω, h(1 − δ_{h})).

This estimate on cylinders may be exploited for balls using the conjugacy assumption.

Proposition 5. Let (f, Λ) be a C^{1+α} conformal expanding map conjugated to a subshift. Then there exists a constant κ ∈ (0, 1) such that for any integer n and any x ∈ Λ

(10) B(x, κ|D_{x}f^{n}|^{−1}) ∩ Λ ⊂ πZ(π^{−1}x, n) ⊂ B(x, κ^{−1}|D_{x}f^{n}|^{−1}) ∩ Λ.

Proof. We give only a sketch. Assume that diam Λ = 1 and |D_{x}f| > 1 for any x ∈ Λ. Since f is conjugated to a subshift the minimal distance between two image πZ and πZ′ of different elements Z, Z′ ∈ Z is bounded from below by some constant δ > 0. Moreover, by distortion there exists a constant D such that for any n and x, y in the same n-cylinder we have |D_{x}f^{n}| ≤ D|D_{y}f^{n}|. Let κ = δ/D.

Suppose that d(x, y) < κ|D_{x}f^{n}|^{−1}. This implies that d(x, y) < δ, thus Z(π^{−1}x, 1) = Z(π^{−1}y, 1). If Z(π^{−1}x, k − 1) = Z(π^{−1}y, k − 1) for some k < n, then we get d(f^{k}x, f^{k}y) ≤ D|D_{x}f^{k}|d(x, y) < δ. Hence Z(π^{−1}x, k) = Z(π^{−1}y, k). By induction we get the first inclusion in Equation (10). The remaining inclusion is easier and we omit its proof. □

Lemma 6. The function γ^{−1} is α-Hölder continuous on g(G_{n}) for any α ∈ (0, 1).

Proof. By arguments similar to those in the proof of Equation (6), but using the continuity of ψ, we get that

(11)  lim_{n→∞} sup_{ω∈G_{n}} |\frac{1}{n}S_{n}ψ(ω) − \frac{1}{n}S_{n}ψ(gω)| = 0.

Let x, y ∈ πG_{n}. Let n be such that

κ exp(−S_{n+1}ψ(gπ^{−1}x)) ≤ d(γx, γy) < κ exp(−S_{n}ψ(gπ^{−1}x)).

It follows from Proposition 5 that gπ^{−1}y ∈ Z(gπ^{−1}x, n). Thus π^{−1}y ∈ Z(π^{−1}x, n(1 − δ_{n})) by (9). Using Proposition 5 again we find that

d(x, y) < κ^{−1} exp(−S_{n(1−δ_{n})}ψ(π^{−1}x)).

For any α ∈ (0, 1), because of (11), the facts that δ_{n} → 0, ψ is strictly positive and bounded, we can find an n(α) such that for all n ≥ n(α) and all x we have

exp(−S_{n(1−δ_{n})}ψ(π^{−1}x)) ≤ κ^{2} exp(−αS_{n+1}ψ(gπ^{−1}x))

This shows that whenever d(γx, γy) < κ inf exp S_{n(α)}ψ we have

d(x, y) < d(γx, γy)^{α},

proving the lemma. □

Since πG_{n} = γ(πgG_{n}) Lemma 6 ensures that the image of the source has large dimension:

(12)  dim_{H} πg(G_{n}) ≥ dim_{H} πG_{n}.

2.5. The image has good recurrence rates. By Proposition 5 we have

(13)  τ_{κ exp(S_{k}ψ(π^{−1}x))}(x) ≥ R_{k}(π^{−1}x) ≥ τ_{κ^{−1} exp(S_{k}ψ(π^{−1}x))}(x),

for any x ∈ Λ. Let λ_{n} = ∫ ψdν_{n}. By Equation (11), g does not change the Birkhoff average of ψ thus for any x ∈ πg(G_{n}), by the property (ii) in the definition of G_{n} we have

(14)  lim_{k→∞} \frac{1}{k} S_{k}ψ(π^{−1}x) = λ.

Then we choose the sequence (ℓ_{k}) as in [3] such that

\underline{lim}_{k→∞} \frac{log ℓ_{k}}{k} = αλ_{n}ν̂_{n}(t) and \overline{lim}_{k→∞} \frac{log ℓ_{k}}{k} = βλ_{n}ν̂_{n}(t).

Using the bounds in (13) and (14) we get that for any x ∈ πg(G_{n})

\underline{lim}_{r→0} \frac{log τ_{r}(x)}{− log r} = \underline{lim}_{k→∞} \frac{log R_{k}(π^{−1}x)}{λ_{n}k} = \underline{lim}_{k→∞} \frac{1}{λ_{n}ν̂_{n}(t)} \frac{log ℓ_{k}}{k} = α,

where the second equality follows from (8). The same arguments for the \overline{lim} gives that x ∈ E(α, β), where

(15) E(α, β) = \{x ∈ Λ: \underline{lim}_{r→0} \frac{log τ_{r}(x)}{− log r} = α and \overline{lim}_{r→0} \frac{log τ_{r}(x)}{− log r} = β\} .

Since πg(G_{n}) ⊂ E(α, β), Theorem 2 follows from (5) and (12).

3. Reduction to a system conjugated to a subshift

We want to ensure that the original system contains subsystems with arbitrary large dimension and which are conjugated to subshifts of finite type. By Bowen’s formula this is the same as finding such subsystems with pressure of the potential − dim_{H}(Λ) log |Df| arbitrarily close to 0. Fernandez, Ugalde and Urias prove that for maps of the interval with a Markov partition by intervals this is always possible [4]. Their proof relies on combinatorial arguments. We provide here a proof based on the spectral approach, close in spirit to the work by Maume and Liverani [9]. Although more intricate, this approach allows us to consider easily higher dimensional conformal maps.

3.1. Pressure for systems with small hole. Let (X, T) be a subshift of finite type. Given an open set H ⊂ X, the hole, we set Y = X \ H and denote by Y^{∞} = ∩_{m≥0}T^{−m}Y the compact forward invariant set of points never falling into the hole H. Our next result says that the pressure of Y^{∞} tends to the pressure of the original system as the hole gets smaller. Notice that this is not a small perturbation for the potential, so that classical continuity of the pressure do not apply.

Proposition 7. Let (X, T) be a topologically mixing subshift of finite type. Assume that φ: X → ℝ is a Hölder continuous potential. Let μ be the equilibrium measure of the potential φ. If H_{n} is a decreasing sequence of ℱ_{n} measurable sets such that for some c, ε > 0

μ(H_{n}) ≤ ce^{−εn}

then lim_{n→∞} P (φ|(X \ H_{n})^{∞}) = P (φ|X).

Proof. The proof uses the characterization of the pressure by the logarithm of the spectral radius of the Perron-Frobenius operator associated to the system. Then we show that one can apply a perturbation result by Keller and Liverani [8] designed for these type of operators. Taking if necessary the new potential φ − P (φ|X) + ψ ∘ T − ψ we can assume that P (φ|X) = 0 and L1 = 1, where L is the Perron-Frobenius operator associated to (X, T, φ), i.e.

Lf(x) = ∑_{y,T y=x} e^{φ(y)}f(y).

Let Z_{n} be the partition of X by n-cylinders, and denote by ℱ_{n} the finite σ-algebra generated by Z_{n}. By the Gibbs property of μ there exists c_{0} such that for any x ∈ Z ∈ Z_{k} we have 1/c_{0} ≤ μ(Z)e^{−φ_{k}(x)} < c_{0}. We furthermore suppose that for any cylinders A, Z such that AZ =^{def} A ∩ T^{−k}Z ≠ ∅. we have 1/c_{0} ≤ μ(A)μ(Z)/μ(AZ) ≤ c_{0}.

Given a function f: X → ℝ and a set A ⊂ X let osc(f, A) = sup_{A} f − inf_{A} f. Since φ is Hölder there exists c_{1} and α < 1 such that we have

(16)  var_{n}(φ) =^{def} sup_{Z∈Z_{n}} osc(φ, Z) ≤ c_{1}α^{n}, for any n ≥ 1.

Fix θ ∈ (1, min(α^{−1}, e^{ε/2})). Define

osc(f) = ∑_{n≥1} θ^{n} ∑_{Z∈Z_{n}} osc(f, Z)μ(Z).

Let |f|_{2} = (∫ |f|^{2}dμ)^{1/2} and consider the Banach space

B = {f: X → ℝ: ∥f∥ =^{def} osc(f) + |f|_{2} < ∞}.

We postpone the proofs of Lemmas 8 and 9 below to Section 4.

Lemma 8. There exists constants c_{3} and c_{4} such that for any k ≥ 1

osc(L^{k}f) ≤ c_{3}θ^{−k} osc(f) + c_{4}|f|_{2}.

Let H_{n}^{k} = H_{n} ∪ T^{−1}H_{n} ∪ ⋯ T^{−k+1}H_{n}. For simplicity we denote by A instead of 1_{A} the indicator function of a set A.

Lemma 9. There exists a constant c_{5} (depending on k) such that for any N ∈ ℕ and f ∈ B,

osc(fH_{N}^{k}) ≤ osc(f) + c_{5}|f|_{2}.

For a set Y ⊂ X we let L_{Y} be the perturbed operator, L_{Y} f = L(Y f). Observe that L_{Y}^{k} f = L^{k}(Y_{k}f), where Y_{k} = Y ∩ T^{−1}Y ∩ ⋯ T^{−k+1}Y. Hence L_{X\H_{N}}^{k} f = L^{k}f − L^{k}(H_{N}^{k}f).

Lemma 10. There exists k and c_{6} such that for any N ∈ ℕ and f ∈ B we have

osc(L_{X\H_{N}}^{k} f) ≤ \frac{1}{2} osc(f) + c_{6}|f|_{2}.

Proof of Lemma 10. Choose k such that c_{3}θ^{−k} ≤ \frac{1}{4}. We apply Lemma 8 and Lemma 9 successively

osc(L_{X\H_{N}}^{k} f) ≤ osc(L^{k}f) + osc(L^{k}(H_{N}^{k}f))

≤ \frac{1}{4} osc(f) + c_{4}|f|_{2} + \frac{1}{4} osc(H_{N}^{k}f) + c_{4}|H_{N}^{k}f|_{2}

≤ \frac{1}{2} osc(f) + (2c_{4} + c_{5})|f|_{2}.

□

The following lemma express that the hole is a perturbation of the original system as far as operators from B to L^{2}(μ) are concerned.

Lemma 11. For any N ∈ ℕ and f ∈ B with ∥f∥ = 1 we have

|(L − L_{X\H_{N}})f|_{2} ≤ c_{0}μ(H_{N})^{\frac{1}{4}} .

Proof of Lemma 11. Using Hölder inequality then the invariance of L we get

∫ |L(H_{N}f)|^{2}dμ ≤ sup_{X} L|f| ∫_{H_{N}} |f|dμ ≤ μ(H_{N})^{\frac{1}{2}} |f|_{2} sup_{X} L|f|.

Additionally, using again Hölder inequality, the supremum is bounded by

sup_{X} L|f| ≤ ∑_{Z∈Z_{1}} c_{0}μ(Z) sup_{Z} |f| ≤ \frac{c_{0}}{θ} osc(f) + c_{0}|f|_{2}.

□

It comes from the Folkolre Theorem that 1 = e^{P (φ|X)} is a simple eigenvalue of L acting on B and that the rest of the spectrum is contained in a disc of radius ρ < 1. By Lemmas 8, 10 and 11 we can apply Keller and Liverani’s theorem ([8], Corollary 1) with the sequence of operators L_{X\H_{n}} acting on B. This gives that this spectral figure will be conserved: for n sufficiently large, the spectrum of L_{X\H_{n}} on B outside the disc of radius ρ consists exactly in one simple eigenvalue λ_{n}, with eigenvector h_{n}, such that λ_{n} → 1 and h_{n} →^{B} h (presumably λ_{n} and h_{n} are positive, but we won’t use this fact).

Furthermore, the restriction of the eigenvector h_{n} to (X \ H_{n})^{∞} is again an eigenvector of the Perron Frobenius operator L_{(X\H_{n})^{∞}} of the system ((X \ H_{n})^{∞}, T). Consequently the spectral radius of L_{(X\H_{n})^{∞}} on B is larger or equal to |λ_{n}|. Thus we get P (φ|X) ≥ P (φ|(X \ H_{n})^{∞}) ≥ log |λ_{n}|, which proves the result. □

3.2. Removing the boundary of the Markov partition. The following proposition is of independent interest.

Proposition 12. Let Λ ⊂ ℳ be a conformal repeller of the C^{1+α} map f: ℳ → ℳ. There exists subsets Λ_{n} ⊂ Λ such that

(i) (f, Λ_{n}) is conjugated to a subshift of finite type

(ii) lim_{n→∞} dim_{H} Λ_{n} = dim_{H} Λ.

Proof. Let ξ be a generating Markov partition for the system (f, Λ) and (X, T) the corresponding subshift of finite type semi-conjugated via π: X → Λ to (f, Λ). We write φ = − dim_{H}(Λ) log |Df|.

Let K = π^{−1}∂ξ be the premiage of the boundary of the partition. The Markov property implies that T K ⊂ K. The compact invariant set K is strictly contained in X hence P (φ|K) < 0 and the ℱ_{n} measurable sets H_{n} = Z_{n}(K) have a measure going exponentially to zero (see for example the appendix of [2] for a proof of these facts). By Proposition 7 for any ϵ > 0 there exists n(ϵ) such that for any n ≥ n(ϵ)

P (φ|(X \ H_{n})^{∞}) > ϵ sup φ,

whence the subset Λ_{n} =^{def} π(X \ H_{n})^{∞} ⊂ Λ has dimension

dim_{H}(Λ_{n}) > (1 − ϵ) dim_{H}(Λ).

Moreover, (f, Λ_{n}) is conjugated to a subshift of finite type. □

We are now able to give a proof of our main theorem.

Proof of Theorem 1. By (i) in Proposition 12 we can apply Theorem 2 to (f, Λ_{n}). This provides us with a set E_{n}(α, β) ⊂ Λ_{n} (See (15)) such that dim E_{n}(α, β) = dim Λ_{n}. By monotony of Hausdorff dimension we get,

dim_{H} Λ ≥ dim_{H} E(α, β) ≥ dim_{H} E_{n}(α, β).

The result follows from (ii) in Proposition 12, taking the limit as n → ∞. □

4. Proofs of the technical lemmas

Proof of Lemma 8. Let φ_{k} = φ + φ ∘ T + ⋯ φ ∘ T^{k−1}. Note that using Equation (16) with c_{2} = c_{1}/(1 − α), for any integer n, k, we have var_{n+k}(φ_{k}) ≤ c_{2}α^{n}. For a cylinder A ∈ Z_{k} and a point x we denote by

RECURRENCE SPECTRUM IN SMOOTH DYNAMICAL SYSTEMS. 11

Ax the unique element of A ∩ T^{−k}x, if any. If Z ∈ Z_{n} with n ≥ 1 then

osc(L^{k}f, Z) = sup_{x,x′∈Z} L^{k}f(x) − L^{k}f(x′)

= sup_{x,x′∈Z} ∑_{A∈Z_{k},AZ≠∅} e^{φ_{k}(Ax)}f(Ax) − e^{φ_{k}(Ax′)}f(Ax′)

≤ sup_{x,x′∈Z} ∑_{A} e^{φ_{k}(Ax)}(f(Ax) − f(Ax′)) + e^{φ_{k}(Ax)}(e^{var_{n+k}(φ_{k})} − 1) |f|(Ax′)

≤ ∑_{A} c_{0}μ(A) osc(f, AZ) + (e^{var_{n+k}(φ_{k})} − 1)c_{0}μ(A) sup_{AZ} |f|

≤ c_{0}(1 + e^{c_{2}}) ∑_{A} μ(A) osc(f, AZ)+

+ c_{0}e^{c_{2}}c_{2}α^{n} ∑_{A} μ(A)μ(AZ)^{−1/2} (∫_{AZ} |f|^{2}dμ)^{1/2} .

Summing up over Z ∈ Z_{n} and then over n yields to

osc(L^{k}f) ≤ ∑_{n≥1} θ^{n} ∑_{Z∈Z_{n+k}}[c_{0}^{2}(1 + e^{c_{2}})μ(Z) osc(f, Z)+
+ c_{0}^{2}e^{c_{2}}c_{2}α^{n}μ(Z)^{1/2} (∫_{Z} |f|^{2}dμ)^{1/2}]
≤ c_{3}θ^{−k} osc(f) + c_{4}|f|_{2}.

For the last inequality we have set c_{3} = c_{0}^{2}(1+e^{c_{2}}) and c_{4} = c_{0}^{2}e^{c_{2}}c_{2}/(1−αθ) and use the Schwarz inequality. □ Proof of Lemma 9. Since H_{N}^{k} is ℱ_{N+k} measurable we have

osc(fH_{N}^{k}) = ∑_{n≥1} θ^{n} ∑_{Z∈Z_{n}} osc(fH_{N}^{k}, Z)μ(Z)

≤ ∑_{n=1}^{N+k−1} θ^{n} ∑_{Z∈Z_{n},Z∩H_{N}^{k} ≠∅} μ(Z) sup_{Z} |f| + ∑_{n≥N+k} θ^{n} ∑_{Z∈Z_{n}} osc(f, Z)μ(Z)

≤ osc(f) + ∑_{n=1}^{N+k−1} θ^{n} ∑_{Z∈Z_{n},Z∩H_{N}^{k} ≠∅} μ(Z)^{1/2} (∫_{Z} |f|^{2}dμ)^{1/2}

≤ osc(f) + ∑_{n=1}^{N+k−1} θ^{n}μ(Z_{n}(H_{N}^{k}))^{1/2}|f|_{2}.

For the last inequality we used Schwarz inequality and Z_{n}(H_{N}^{k}) denotes the union of elements of Z_{n} intersecting H_{N}^{k}. For any n ≤ N + k − 1 we have

Z_{n}(H_{N}^{k}) ⊂ ∪_{j=0}^{k−1}T^{−j}Z_{n−j}(H_{N}) ⊂ ∪_{j=0}^{k−1}T^{−j}H_{n−k}.

The result follows from the invariance of the measure by taking

c_{5} = \frac{\sqrt{cke^{εk}}}{1 − θe^{− \frac{ε}{2}}} ≥ sup_{N} ∑_{n=1}^{N+k−1} θ^{n}(kce^{−ε(n−k)})^{1/2}.

□

Acknowledgments. The first author would like to thank Luis Barreira and Roman Hric for interesting discussions on the topic and the organizers of the meeting “Recent trends in Dynamics 2002” held in Porto, Portugal. The second author would like to thank CNRS UMR 6140, Amiens, where he encountered warm hospitality.

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CNRS UMR 6140 - LAMFA, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France

E-mail address: benoit.saussol@u-picardie.fr
URL: http://www.mathinfo.u-picardie.fr/saussol/

Department of Mathematics and Nonlinear Science Center, Wuhan University, Wuhan, 430072, People’s Republic of China

E-mail address: wujunyu@public.wh.hb.cn