LARGE DEVIATIONS FOR RETURN TIMES IN

NON-RECTANGLE SETS FOR AXIOM A DIFFEOMORPHISMS

Renaud Leplaideur

Benoît Saussol

Laboratoire de Mathématiques, CNRS UMR 6205, Université de Bretagne Occidentale, 6 av. Victor Le Gorgeu CS 93837, 29238 BREST Cedex 3, FRANCE

(Communicated by the associate editor name)

Abstract. For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.

1. Introduction

Recall that for any given measurable and ergodic dynamical system (X, T, μ), and for any set A with positive μ-measure, Kač ’s lemma together with birkhoff ergodic theorem implies that the sequence r_{A}^{n} of nth return-times into A by iterations of the map T satisfies

lim_{n→∞} \frac{r_{A}^{n}(x)}{n} = \frac{1}{μ(A)} for μ-a.e. x.

We are interested in fluctuations of order n of r_{A}^{n} around \frac{n}{μ(A)}, so we want to prove a Large Deviations Principle (LDP for short), that is, we want to show the existence of the rate function Φ_{A} such that for every u ≥ \frac{1}{μ(A)},

lim_{n→∞} \frac{1}{n} log μ \{\frac{r_{A}^{n}}{n} ≥ u\} = Φ_{A}(u)

and for every 0 ≤ u ≤ \frac{1}{μ(A)},

lim_{n→∞} \frac{1}{n} log μ \{\frac{r_{A}^{n}}{n} ≤ u\} = Φ_{A}(u).

If this holds, we will say that the sequence of return-times into the set A satisfies the LDP for the measure μ. If the above LDP only holds for u ⊂]\underline{u}, \overline{u}[ with \underline{u} < \frac{1}{μ(A)} < \overline{u}, we will say that the sequence of return-times into A satisfies a LDP for the measure μ near the average.

A standard method to get a LDP is to prove the existence of the scaled-cumulant generating function Ψ_{A}, defined by the following limit

Ψ_{A}(α) = lim_{n→∞} \frac{1}{n} log ∫ e^{αr_{A}^{n}}dμ,

Date: December 2007.

2000 Mathematics Subject Classification. Primary: 37B20, 37C45, 37D20, 37D35, 60F10.

Key words and phrases. large deviations, return times, thermodynamic formalism.

and prove that it is differentiable. In this case it is well known that the rate function exists and these two functions form a Legendre transform pair, namely

(1)  Φ_{A}(u) = inf_{α} {Ψ_{A}(α) − αu} .

This is the approach that we are adopting in this paper, with the difference that we do not prove the differentiability of the function Ψ_{A}. It is noteworthy that with our monotone approximation method this assumption is not required (see Proposition 5 for details).

Our result applies to Axiom A diffeomorphism and equilibrium state of Hölder potential, and for the sequence of successive return times into a Borel set A that satisfies a condition about the smallness of its (non-Markovian) boundary. Taking a Markov partition and the corresponding semi-conjugacy, we will state the result for subshifts of finite type, keeping in mind that this really corresponds to a result for Borel sets on the manifold.

2. Statements

Throughout, (Σ, σ) will denote a topologically mixing subshift of finite type. The set of vertices of the defining graph of (Σ, σ) is {1, …, N} with N ≥ 2. We denote by A = (a_{ij}) the N ×N-transition (irreducible and aperiodic) matrix associated to Σ; namely points in Σ are sequences x = {x_{n}}_{n∈ℤ} such that for every n, x_{n} belongs to {1, …, N} and

a_{x_{n}x_{n+1}} = 1.

Recall that if f is an Axiom A diffeomorphism of a compact manifold M then there always exists a subshift of finite type Σ and a coding map π: Σ → M such that π ∘ σ = f ∘ π.

Let ϕ : Σ → ℝ be α-Hölder continuous. For a given σ-invariant measure λ, the ϕ-pressure is the quantity P_{λ}(ϕ) := h_{λ}(σ) + ∫ ϕ dλ; P_{λ}(ϕ) will also be called the λ-pressure of ϕ. The unique equilibrium state for ϕ, i.e. the unique σ-invariant probability measure with maximal ϕ-pressure, will be denoted by μ_{ϕ}. Its pressure is the topological ϕ-pressure.

For a set A ⊂ Σ and an integer n, we denote by ∂A its topological boundary \overline{A} ∩ \overline{Σ \ A}.

Note that ∂A can be empty; this holds for example when A is a finite union of cylinders. We let P̃_{ϕ}(∂A) be the ϕ pressure of ∂A ; since ∂A may not be invariant we define it according to the variational principle:

P̃_{ϕ}(∂A) = sup {h_{ν}(σ) + ∫ ϕdν: ν ergodic and ν(∂A) > 0}

Note that this does not correspond to the dimension-like definition of the pressure introduced by Pesin and Pitskel [7].

If D is any subset in Σ, and for x ∈ Σ, we denote by r_{D}(x) the first hitting in D by iterations of σ (if it exists). Namely r_{D}(x) is the smallest integer n > 1 such that σ^{n}(x) belongs to D, and r_{D}(x) = +∞ if no such integer exists. We also set r_{D}^{1}(x) = r_{D}(x), and denote the nth return time r_{D}^{n}(x) the cocycle defined by

r_{D}^{n+1}(x) = r_{D}^{n}(x) + r_{D}(σ^{r_{D}^{n}(x)}(x)).

Let max(A) := sup_{μ} μ(A) and min(A) = inf_{μ} μ(A), where the extrema are taken among all invariant measures μ. Then our result is:

Theorem. Let ϕ : Σ → ℝ be any Hölder continuous function. Let A ⊂ Σ be a Borel set. We have:

(1) if for any σ-invariant measure μ, μ(∂A) = 0, then the sequence (r_{A}^{n})_{n≥1} satisfies the Large Deviations Principle for μ_{ϕ}, except possibly for discontinuity points. More precisely, for any u ∈ (\frac{1}{max(A)}, \frac{1}{min(A)}) the rate function

Φ_{A}(u) exists and is finite, and for any u outside the interval [\frac{1}{max(A)}, \frac{1}{min(A)}], the rate function Φ_{A}(u) exists is equal to −∞.

(2) if the ϕ-pressure P̃_{ϕ}(∂A) of the boundary is strictly smaller than the ϕtopological pressure then the sequence (r_{A}^{n})_{n≥1} satisfies a Large Deviations Principle for μ_{ϕ} near the average.

We recall that our method is based on the existence of the cumulant generating function Ψ_{A}(α) for every α in some open interval (\underline{α}, \overline{α}). For the statement 1, we will prove that the function Ψ_{A} is defined on an interval (−∞, \overline{α}). For the statement 2, we will only get the existence of Ψ_{A} on some open neighborhood (\underline{α}, \overline{α}) of 0.

We emphasize that if f is an axiom A diffeomorphism of a manifold M and μ_{ϕ} is an equilibrium state of a Hölder potential ϕ, and V is a Borel set then the theorem applies to A = π^{−1}V since ∂π^{−1}V ⊂ π^{−1}∂V under the same hypotheses on ∂V. The hypothesis in statement 1 could seem very restrictive; however it should be satisfied quite often in some general situations, as the following example suggests.

Example 1. Let (M, f) be an hyperbolic automorphism of the 2-torus, and consider the family of balls B(x, r) about a given point x ∈ M. Then, for all but countably many radii r > 0, the condition μ(∂B(x, r)) = 0 for every invariant measure μ is satisfied.

Proof. Let S be the boundary of a ball. Using hyperbolicity one can show that the intersection of S with its images f^{n}(S) consists, at most, of countably many points. Hence the set of recurrent points R(S) in S is at most countable. If an invariant measure gives weight to S, by Poincaré Recurrence Theorem it implies that it gives weight to R(S) which is a countable set. Thus the measure must have an atomic part consisting of a periodic orbit. Hence S must contain a periodic point. Since the set of periodic points of such a map is countable there can be at most countably many boundaries ∂B(x, r) carrying a periodic point as r varies, which proves the proposition. □

The hypothesis in statement 2 about the pressure of the boundary appears quite naturally in the thermodynamic formalism of dynamical systems with singularities. In the case of ϕ = 0 it simply says that the boundary ∂A does not carry full measure theoretical entropy. A more explicit condition can be given on the manifold itself.

Proposition 1. Let f be an axiom A diffeomorphism of a manifold M and let μ_{ϕ} be an equilibrium state of a Hölder potential ϕ. Let V be a Borel set and denote by U_{ε}(∂V) the ε-neighborhood of the boundary ∂V. Assume that there exist some constants c > 0 and a > 0 such that

μ_{ϕ}(U_{ε}(∂V)) ≤ cε^{a} ∀ε > 0.

Then, P̃_{ϕ}(∂V) < P_{ϕ}(M). In particular the sequence of return times into V satisfies a Large Deviations Principle near the average.

See Section 5 for further details. In particular we have:

Example 2. Let (M, f) be a C^{2} volume preserving Anosov diffeomorphism and let V ⊂ M be an Borel set with piecewise C^{1} boundary. Then the sequence of return times into V satisfies a Large Deviations Principle near the average.

Proof. Set ϕ = − log |D_{x}^{u}f|. The equilibrium measure μ_{ϕ} is the SBR measure which is here the volume measure, and the assumption in Proposition 1 is clearly satisfied. □

Outline of the proof of the theorem: in Section 3 we recall how the LDP was obtained for the return-times in cylinders. In Section 4 we compare the cumulant generating functions of inner and outer approximation of our set A by union of cylinders. In section 5, under the assumption of the statement 2 of the theorem, we prove the existence of the cumulant generating function Ψ_{A} on some interval. In section 6 we give a dynamical proof of the first statement of the theorem.

3. Large deviations for return time in cylinders

We first recall the local thermodynamic formalism introduced in [4]. Then we recall how the large deviations principle for union of cylinders was obtained in [2]. Finally, we derive a uniform mass concentration principle.

3.1. Induced systems and local thermodynamic formalism. For a given point x = (x_{n})_{n∈ℤ} ∈ Σ, the past (resp. future) of the point denotes the backward (resp. forward) sequence (x_{n})_{n≤0} (resp. (x_{n})_{n≥0}). For x and y in Σ, when x_{0} = y_{0}, the point z := [x, y] is the point obtained when we take the past of y and the future of x.

In Σ, the cylinder [i_{k}, …, i_{k+n}] will denote the set of points x ∈ Σ such that x_{j} = i_{j} (for every k ≤ j ≤ k + n). Such a cylinder will also be called a word (of length n + 1) or equivalently a (k, k + n)-cylinder. If x is in Σ, C_{k,k+n}(x) will denote the cylinder [i_{k}, …, i_{k+n}] such that x_{j} = i_{j} (for every k ≤ j ≤ k + n). By extension, C_{−∞,n}(x) will denotes the set of points (y_{k}) such that y_{k} = x_{k} for every k ≤ n; similarly C_{n,+∞}(x) will denotes the set of points (y_{k}) such that y_{k} = x_{k} for every k ≥ n. By definition, the local unstable leaf W_{loc}^{u}(x) is C_{−∞,0}(x), and the local stable leaf W_{loc}^{s}(x) is C_{0,+∞}(x). For n ≥ 0, a n-cylinder will denote a (−n, n)- cylinder. The letter R = ∪R_{i} denotes some finite union of (−n, n)-cylinders; in each of these cylinders we fix some local unstable leaf F_{i}. There is a natural projection from each R_{i} onto each F_{i} defined by π_{F_{i}}(z) = [z, x], where x is any point in F_{i}. For convenience we denote by π_{F} the map defined on R by

π_{F} (z) = π_{F_{i}}(z) ⟺ z ∈ R_{i}.

We denote by g the first return map in R, and by g_{F} the map π_{F} ∘ g. We thus have g(x) = σ^{r_{R}(x)}(x). Note that if the maps r_{R}, g and g_{F} are not defined everywhere in R, the inverse branches of g_{F} are well defined in the whole F.

We can thus define the Ruelle-Perron-Frobenius operator for g_{F}: for x in F, we set

L_{S}(T )(x) = ∑_{y, g_{F} (y)=x} e^{S_{r_{R}(y)}(ϕ)(y)−r_{R}(y)S}T (y),

where T : F → ℝ is a continuous function, and S is a real parameter. As usual, S_{n}(ϕ)(x) denotes the Birkhoff sum ϕ(x) + ⋯ + ϕ ∘ σ^{n−1}(x).

There exists some critical S_{c}, such that for every S > S_{c} all the following holds: L_{S} admits a unique and single dominating eigenvalue λ_{S} in the set of α-Hölder continuous functions. The adjoint operator L_{S}^{∗} has also λ_{S} for unique and single dominating eigenvalue; we denote by ν_{S} the unique probability measure on F such that L_{S}^{∗}(ν_{S}) = λ_{S}ν_{S}. We denote by H_{S}, the unique α-Hölder continuous and positive function on F satisfying L_{S}(H_{S}) = λ_{S}H_{S} and ∫ H_{S} dν_{S} = 1. We also denote by μ_{S} the measure H_{S}ν_{S}, and by μ̂_{S} the natural extension of μ_{S}. We recall that μ_{S} is a g_{F} -invariant probability measure, and μ̂_{S} is a g-invariant probability measure. At last, we denote by m_{S} the opened-out measure: namely m_{S} is the σ-invariant measure satisfying, m_{S}(R) > 0, and μ̂_{S} is the conditional measure m_{S}(·|R).

The spectral properties of L_{S} yield the existence of positive real constants C_{ϕ} and ε_{S}, such that for every Hölder continuous T : F → ℝ, for every integer n ≥ 1 and for every x in F

(2)  L_{S}^{n}(T )(x) = e^{n log λ_{S}} ∫ T dν_{S}H_{S}(x) + O(e^{n(log λ_{S}−ε_{S})})||T ||_{∞}.

Note that H_{S} is a positive function on the compact set F.

In [2], it is proved that the critical value S_{c} is the pressure of the dotted system, with hole R, associated to the potential ϕ. Namely we consider in Σ the system Σ_{R} := ⋂_{n∈ℤ} σ^{−n}(Σ \ R). Although not explicitly mentioned, the case Σ_{R} = ∅ appears, when min(R) ≠ 0. In this case one simply has S_{c} = −∞ and the identity remains valid with the convention that the pressure of the emptyset is −∞. The proof in [2] was done under the assumption that Σ_{R} is mixing. We claim that the mixing hypothesis can be omitted. Indeed, any subshift of finite type can be decomposed in irreducible components, which satisfy the mixing property, but for some iteration of the map σ (see e.g. [1]). As we are considering first returns in R, note that the word defined by the cylinder C_{0,r_{R}(x)}(x) contains no R_{i} but at the first position. Now, two different irreducible components can be joined in Σ only by a path which contains R. Therefore, the word defined by the cylinder C_{0,r_{R}(x)}(x) is an admissible word for a unique irreducible component.

Unicity of the equilibrium state in any mixing subshift (for ϕ) implies that the topological ϕ-pressure P_{ϕ}(Σ_{R}) for (Σ_{R}, σ) is strictly lower than the topological ϕpressure for Σ, P_{ϕ}(Σ).

We now finish this subsection with some important characterization for the measure m_{S}.

Lemma 3.1. The measure m_{S} is the unique equilibrium state in (Σ, σ) associated to ϕ − log λ_{S}1_{R}. Moreover, its ϕ − log λ_{S}1_{R}-pressure is S.

Proof. For simplicity we set β := log λ_{S}. The measure m_{S} satisfies,

(3)  h_{m_{S}}(σ) + ∫ ϕ dm_{S} = S + m_{S}(R)β.

We refer the reader to [4], Proposition 6.8 for a proof. Moreover, the measure μ̂_{S} is the unique equilibrium state for (R, g) associated to the potential S_{r(·)}(ϕ)(·)−Sr(·), with pressure β. Let us pick some σ-invariant probability measure ν.

Let us first assume that ν(R) > 0. We have

h_{ν}(σ) + ∫ ϕ dν − S = ν(R) (h_{ν_{|R}}(g) + ∫ S_{r(·)}(ϕ) dν_{|R} − S ∫ r(·) dν_{|R}) ,
≤ ν(R)β,

where ν_{|R} is the conditional measure ν(·|R). This gives

h_{ν}(σ) + ∫ ϕ dν − β ∫ 1_{R} dν ≤ S,

with equality if and only if ν_{|R} = μ̂_{S} (i.e. m_{S} = ν).

If we assume that ν(R) = 0, then ν is a σ-invariant probability measure with support in Σ_{R}. Therefore it must satisfy

h_{ν}(σ) + ∫ ϕ dν − β ∫ 1_{R} dν = h_{ν}(σ) + ∫ ϕ dν ≤ S_{c} < S.

This finishes the proof of the lemma. □

3.2. Large deviations for return times in cylinders. In [2], it is also proved that λ_{S} → +∞ as S goes to S_{c}. Moreover, the map S ↦ log λ_{S} is a decreasing convex map on ]S_{c}, +∞[. There also exists some complex neighborhood of ]S_{c}, +∞[ such that the map S ↦ log λ_{S} admits an analytic continuation on it. In particular the map S ↦ log λ_{S} is real-analytic on ]S_{c}, +∞[.

Finally, it is proved in [2] that for every α < α(R) := P_{ϕ}(Σ) − P_{ϕ}(Σ_{R}),

(4)  lim_{n→∞} \frac{1}{n} log ∫_{R} e^{αr_{R}^{n}(x)} dμ_{ϕ} = log λ_{P_{ϕ}(Σ)−α}.

We shall show now that the large deviations for successive return time and entrance time is the same question; namely, the fact that we are starting from the set R or from the whole space to compute the integral does not make any difference.

Proposition 2. If R and S are non-empty finite unions of cylinders, then

Ψ_{R}(α) = lim \frac{1}{n} log ∫_{S} e^{αr_{R}^{n}}dμ_{ϕ},

in particular we have Ψ_{R}(α) = log λ_{P_{ϕ}(Σ)−α}.

Remark 1. As mentioned in the introduction, this readily implies the large deviations principle for return times in the form given by (1) since the function Ψ_{R} is differentiable.

The proposition is a weak consequence of the ψ-mixing property of the measure μ_{ϕ}. Indeed, there exists M > 0 and κ > 1 such that if f and g are two integrable functions such that f(x) only depends on (x_{n})_{n≤p} and g(x) only depends on (x_{n})_{n≥p+M} then

(5)  κ^{−1} ∫ fdμ_{ϕ} ∫ gdμ_{ϕ} ≤ ∫ fgdμ_{ϕ} ≤ κ ∫ fdμ_{ϕ} ∫ gdμ_{ϕ}.

Lemma 3.2. If R and S are finite union of (−m, m) cylinder then for any n ≥ M + 2m and for

\begin{aligned}κ^{−1}μ(S) ∫_{Σ} e^{αr_{R}^{n−(M+2m)}}dμ_{ϕ} & ≤ ∫_{S} e^{αr_{R}^{n}}dμ_{ϕ} ≤ κμ(S)e^{α(M+2m)} ∫_{Σ} e^{αr_{R}^{n}}dμ_{ϕ} (α ≥ 0) \\ κ^{−1}μ(S)e^{α(M+2m)} ∫_{Σ} e^{αr_{R}^{n}}dμ_{ϕ} & ≤ ∫_{S} e^{αr_{R}^{n}}dμ_{ϕ} ≤ κμ(S) ∫_{Σ} e^{αr_{R}^{n−(M+2m)}}dμ_{ϕ} (α ≤ 0). \end{aligned}

Proof. For any n ≥ M + 2m we have

r_{R}^{n−(M+2m)} ∘ f^{M+2m} ≤ r_{R}^{n} ≤ M + 2m + r_{R}^{n} ∘ f^{M+2m}

from which the result follows by inequality (5). □

The proof of the proposition consists in applying twice the lemma : from the integration over S to Σ and then to R.

3.3. Concentration of the mass. Let R be a finite union of cylinders. The large deviations principle holds for the return times r_{R}^{n}. It is well-known that this implies a kind of concentration of the mass.

Proposition 3. Let R be a finite union of cylinders. Let α and δ > 0 such that

α + δ < α(R). Then for every τ > \frac{Ψ_{R}(α + δ) − Ψ_{R}(δ)}{δ} we have

Ψ_{R}(α) = lim_{n→∞} \frac{1}{n} log ∫ e^{αr_{R}^{n}}dμ_{ϕ} = lim_{n→∞} \frac{1}{n} log ∫_{\{r_{R}^{n}≤nτ\}} e^{αr_{R}^{n}}dμ_{ϕ}.

Proof. Take ε > 0 so small that −δτ + Ψ_{R}(α + δ) + ε ≤ Ψ_{R}(α) − ε. Using Markov inequality we get

∫_{r_{R}^{n}>nτ} e^{αr_{R}^{n}}dμ_{ϕ} = ∫_{r_{R}^{n}>nτ} e^{(α+δ)r_{R}^{n}}e^{−δr_{R}^{n}}dμ_{ϕ}
≤ e^{−δnτ} ∫_{r_{R}^{n}>nτ} e^{(α+δ)r_{R}^{n}}dμ_{ϕ}
≤ e^{−δnτ} ∫ e^{(α+δ)r_{R}^{n}}dμ_{ϕ}
≤ e^{−δnτ} e^{n(Ψ_{R}(α+δ)+ε)}
= o(e^{nΨ_{R}(α)})
□

4. Existence of inner and outer approximations, their properties and consequences of their equality

In the first subsection we prove a monotonicity result about cumulant generating functions. The idea is to approximate the set A from the inside and from the outside by finite unions of cylinders, and show that the inner cumulant generating function Ψ_{in} and the outer cumulant generating function Ψ_{out} exist. Finally, we study the consequence of their equality on the cumulant generating function and the rate function for the set A.

4.1. Monotonicity of the cumulant generating function on rectangles. For m an integer, let B_{m} be the biggest union of m-cylinders contained in A and C_{m} be the smallest union of m-cylinders which contains A. Then, we denote by D_{m} the set C_{m} \ B_{m} (See Figure 1). As any (−m, m)-cylinder is a union of (−m − 1, m + 1)- cylinders, we have B_{m} ⊂ B_{m+1} ⊂ A ⊂ C_{m+1} ⊂ C_{m}; Therefore (D_{m}) is a decreasing sequence of compact set which converges to ∂A.

Following what is done above, there exists two analytic functions Ψ_{B_{m}} and Ψ_{C_{m}} respectively defined on ] − ∞, α(B_{m})[ and ] − ∞, α(C_{m})[. As it is said above, α(B_{m}) is the difference between P_{ϕ}(Σ) and the topological ϕ-pressure of the dotted system Σ_{B_{m}}. In the same way, α(C_{m}) is the difference between P_{ϕ}(Σ) and the topological ϕpressure of the dotted system Σ_{C_{m}}. Now, we clearly have Σ_{C_{m}} ⊂ Σ_{B_{m}}, because C_{m} ⊃ B_{m}. We also have C_{m+1} ⊂ C_{m} and B_{m} ⊂ B_{m+1}. Therefore the sequence (α(B_{m}))_{m} is non-decreasing and the sequence (α(C_{m}))_{m} is non-increasing. Moreover, for any m,

α(B_{m}) ≤ α(C_{m}).

B_{m}  A  C_{m} D_{m}

Figure 1. Inner and outer approximation of the set A by m-cylinders.

The sequence (α(B_{m}))_{m} is thus converging to some limit α_{in} ∈ (0, +∞]. Hence, for any α < α_{in}, and for any sufficiently large m, the functions Ψ_{B_{m}} and Ψ_{C_{m}} are real-analytic on ] − ∞, α[.

Let us define the lower and upper cumulant generating functions of A by

Ψ_{A}(α) = lim inf_{n→∞} {1 \atop n} log ∫ e^{αr_{A}^{n}}dμ_{ϕ} and Ψ_{A}(α) = lim sup_{n→∞} {1 \atop n} log ∫ e^{αr_{A}^{n}}dμ_{ϕ}.

Proposition 4. For any 0 ≤ α < α_{in} and for any sufficiently large m, we have

Ψ_{C_{m}}(α) ≤ Ψ_{A}(α) ≤ Ψ_{A}(α) ≤ Ψ_{B_{m}}(α).

For any α < 0 we have

Ψ_{B_{m}}(α) ≤ Ψ_{A}(α) ≤ Ψ_{A}(α) ≤ Ψ_{C_{m}}(α).

Proof. The double inclusion B_{m} ⊂ A ⊂ C_{m} implies that r_{B_{m}}^{n} ≥ r_{A}^{n} ≥ r_{C_{m}}^{n}. The result follows then immediately by integration and taking the appropriate limits. □

Remark 2. Forgetting the set A in the previous proof, we have in fact proved that if B and C are finite unions of cylinders satisfying B ⊂ C, then for any α < 0,

(6)  Ψ_{B}(α) ≤ Ψ_{C}(α),

and for any α ≥ 0 (but sufficiently small such that the functions are well-defined)

(7)  Ψ_{B}(α) ≥ Ψ_{C}(α).

4.2. Existence of inner and outer cumulant generating functions. Let us pick some 0 < α < α_{in}. By (7), the sequence of functions (Ψ_{B_{m}})_{m} is a non-increasing sequence of non-decreasing convex functions on [0, α[ (for sufficiently large m !). It thus (simply) converges to some limit function Ψ_{in}. This function Ψ_{in} has to be convex, thus continuous on ]0, α[. It also has to be non-decreasing, thus it must be continuous on [0, α[. Moreover the Dini Theorem yields that the convergence is uniform on every compact set included in [0, α[. This occurs for any 0 < α < α_{in}, thus the limit function Ψ_{in} is non-decreasing and continuous on [0, α_{in}[ and the convergence is uniform on every compact set included in [0, α_{in}[.

In the same way the sequence of functions (Ψ_{C_{m}})_{m} is a non-decreasing sequence of non-decreasing convex functions on [0, α_{in}[ (for any m !). It thus (simply) converges to some limit function Ψ_{out}; this function Ψ_{out} is convex and continuous on ]0, α_{in}[. Note that by (7) we have

0 ≤ Ψ_{out} ≤ Ψ_{in}.

As Ψ_{B_{m}}(0) = Ψ_{C_{m}}(0) = 0 for any m, Ψ_{out} is continuous on [0, α_{in}[, and the convergence is uniform.

We do the same work on ] − ∞, 0] using (6) instead of (7). Note that for α ≤ 0 we have

0 ≥ Ψ_{out}(α) ≥ Ψ_{in}(α).

The two functions Ψ_{in} and Ψ_{out} are convex and non-decreasing. By Proposition 4 we have

Ψ_{out}(α) ≤ \underline{Ψ}_{A}(α) ≤ \overline{Ψ}_{A}(α) ≤ Ψ_{in}(α) for α ≥ 0,
and Ψ_{in}(α) ≤ \underline{Ψ}_{A}(α) ≤ \overline{Ψ}_{A}(α) ≤ Ψ_{out}(α) for α ≤ 0.

We emphasize that the existence of the limit Ψ_{A}(α) immediately follows from the equality Ψ_{in}(α) = Ψ_{out}(α). Moreover, it also implies, quite surprisingly, despite any knowledge about the differentiability of the function Ψ_{A}, that Φ_{A} exists and is the Legendre transform of Ψ_{A}:

Proposition 5. If Ψ_{in} = Ψ_{out} on an open interval (\underline{α}, \overline{α}) then

(i) the cumulant generating function Ψ_{A} exists, is equal to Ψ_{in} = Ψ_{out} on this interval and it is a convex, continuous, non-decreasing function. In addition Ψ_{A} is differentiable for all but countably many points and left and right limits of Ψ′_{A} exists everywhere on the closure of the interval.

(ii) for all u ∈ (Ψ′_{A}(\underline{α}+), Ψ′_{A}(\overline{α}−)), the rate function Φ_{A}(u) exists and satisfies the relation

Φ_{A}(u) = inf_{α∈(\underline{α},\overline{α})}\{Ψ_{A}(α) − αu\}.

(iii) if \underline{α} = −∞ then for any u < Ψ′_{A}(\overline{α}), except possibly for u = Ψ′_{A}(−∞), the rate function Φ_{A}(u) exists and satisfies the relation

Φ_{A}(u) = inf_{α<\overline{α}}\{Ψ_{A}(α) − αu\}.

Proof. (i) is straightforward.

(ii) Using exponential Markov inequality we immediately get the upper bound

\overline{Φ}_{A}(u) := lim sup_{n→∞} \frac{1}{n} log μ_{ϕ}(r_{A}^{n} ≥ nu) ≤ inf_{α} Ψ_{A}(α) − αu (u > \frac{1}{μ}(A))
\overline{Φ}_{A}(u) := lim sup_{n→∞} \frac{1}{n} log μ_{ϕ}(r_{A}^{n} ≤ nu) ≤ inf_{α} Ψ_{A}(α) − αu (u < \frac{1}{μ}(A)).

(8)

We now prove that for \underline{Φ}_{A} defined with the lim inf the lower bound also holds true. Fix u ∈ (Ψ′_{A}(\underline{α}+), Ψ′_{A}(\overline{α}−)). By convexity of Ψ_{A} the function Ψ_{A}(α) − αu attains its infimum for some α_{∗} ∈ (\underline{α}, \overline{α}), and the limits at the endpoints of the interval are strictly larger. Let ε > 0 so small that there exists \underline{β} < α_{∗} < \overline{β} in the interval (\underline{α}, \overline{α}) such that for α = \underline{β} and \overline{β} we have Ψ_{A}(α) − αu > Ψ_{A}(α_{∗}) − α_{∗}u + 2ε.

Suppose α_{∗} > 0, i.e. u > \frac{1}{μ(A)}. By equality of the outer approximation Ψ_{out} with Ψ_{A}, there exists a set C which is a finite union of cylinders such that A ⊂ C and Ψ_{C} ≤ Ψ_{A} ≤ Ψ_{C} + ε on the interval [0, \overline{β}]. This implies that Ψ_{C}(\overline{β}) − \overline{β}u > Ψ_{C}(α_{∗}) − α_{∗}u, therefore by convexity of Ψ_{C}(α) − αu we have

(9)  inf_{α} Ψ_{C}(α) − αu = inf_{α∈[0,\overline{β}]} Ψ_{C}(α) − αu ≥ inf_{α} Ψ_{A}(α) − αu − ε.

On the other hand r_{C}^{n} ≥ nu implies r_{A}^{n} ≥ nu, hence \underline{Φ}_{A}(u) ≥ Φ_{C}(u). Since the large deviations principle holds for the set C (see Remark 1), we have Φ_{C}(u) = inf_{α} Ψ_{C}(α) − α and the conclusion follows from inequality (9). The case α_{∗} < 0 can be treated in the same way by considering the inner approximation Ψ_{in}.

(iii) By (ii) it is enough to consider u < lim_{α→−∞} Ψ′_{A}(α), but in this case inf_{α} Ψ_{A}(u) − αu = −∞ which implies the result by (8). □

We remark that the exceptional value Ψ′(−∞) in statement (iii) is the slope of Ψ_{A} at −∞ which is \frac{1}{max(A)}. At this point the value of Φ_{A} jumps from −∞ to finite values.

Note that if lim_{α→\overline{α}} Ψ′_{A}(α) = +∞ in Proposition 5 then one gets the Large Deviations Principle and the formula (1) holds for every u > Ψ′_{A}(\underline{α}+). In Sections 6 we will prove under the assumption in statement 1 of the theorem the existence of Ψ_{A} on the interval (−∞, α_{in}). However, we are not able to prove that the limit of the derivative at α_{in} is infinite. We doubt that it could be finite when α_{in} < +∞. Still, this is clearly the case whenever α_{in} = +∞, which is equivalent to min(A) > 0.

Nevertheless, using next proposition which exploits the symmetry of our assumption (since A and A^{c} share the same boundary), this will be sufficient to get the existence of the rate function Φ_{A} on the whole interval (except at the discontinuity).

Proposition 6. Assume that the Large Deviations Principle of return times into A^{c} holds for any u ≤ \frac{1}{μ(A^{c})} with a continuous rate function Φ_{A^{c}} (except possibly for u = \frac{1}{max(A^{c})}). Then the Large Deviations Principle for return times into A holds for any u ≥ \frac{1}{μ(A)} (except possibly for u = \frac{1}{min(A)}) with a rate function Φ_{A} which satisfies

Φ_{A}(u) = (u − 1) Φ_{A^{c}} (\frac{u}{u − 1}) .

In particular, if Φ_{A^{c}} is the Legendre transform of the cumulant generating function Ψ_{A^{c}} then

Φ_{A}(u) = inf_{α<0} {−αu + (u − 1)Ψ_{A^{c}} (α)} .

Proof. Observe that if u > \frac{1}{μ(A)} ≥ 1 then r_{A}^{n} ≥ nu if and only if the orbit entered at most n times in A before the time ⌊nu⌋, which means that the orbit entered at least ⌊n(u − 1)⌋ times into A^{c} before the time ⌊nu⌋. Therefore

\frac{1}{n} log μ_{ϕ}(r_{A}^{n} ≥ nu) = \frac{1}{n} log μ_{ϕ} (r_{A^{c}}^{⌊n(u−1)⌋} < ⌊n(u − 1)⌋\frac{⌊nu⌋}{⌊n(u − 1)⌋})

and the result follows by taking the limit as n → ∞.

□

5. Coincidence of inner and outer approximation in the case of a small pressure boundary

The goal of this section is to prove the statement 2 of the theorem. By the previous analysis (See Proposition 5) it is sufficient to prove the existence of some interval (\underline{α}, \overline{α}) ∋ 0 such that for any α ∈ (\underline{α}, \overline{α}) we have Ψ_{in}(α) = Ψ_{out}(α).

5.1. A more explicit condition to get a small pressure boundary. Let K ⊂ Σ be a Borel set. We recall our definition of its ϕ-pressure:

P̃_{ϕ}(K) = sup {h_{ν} + ∫ ϕdν: ν ergodic and ν(K) > 0} .

Note that it is not the same as the one defined as a dimension like characteristic with forward cylinders. That would satisfy us in the case of expanding maps, but for diffeomorphisms it would lead to a condition much too strong.

Proposition 7. Let K ⊂ Σ be a Borel set and let V_{n}(K) be the smallest union of (−n, n)-cylinders which contains K. If there exist some constants c > 0 and θ > 0 such that μ_{ϕ}(V_{n}(K)) ≤ ce^{−θn} for all integers n, then P̃_{ϕ}(K) ≤ P_{ϕ}(Σ) − \frac{1}{2}θ.

Proof. Let S̃_{n}ϕ(x) = ∑_{k=−n}^{n−1} ϕ(x) denote the two sided Birkhoff sum and C_{−n,n}(x) the (−n, n)-cylinder containing the point x ∈ Σ. Recall that since μ_{ϕ} is a Gibbs measure, for some constant b > 0 and for every x ∈ Σ we have

e^{−b} ≤ \frac{μ_{ϕ}(C_{−n,n}(x))}{exp (S̃_{n}ϕ(x) − 2nP_{ϕ}(Σ))} ≤ e^{b}.

Let ν be an ergodic measure such that ν(K) > 0. We have

c ≥ e^{θn}μ_{ϕ}(V_{n}(K))

≥ ∫_{K} exp (θn + log \frac{μ_{ϕ}(C_{−n,n}(x))}{ν(C_{−n,n}(x))}) dν(x)

= ∫_{K} exp (2n [\frac{θ}{2} + P_{ϕ}(Σ) + \frac{1}{2n}S̃_{n}ϕ(x) − \frac{1}{2n} log ν(C_{−n,n}(x))]) dν(x).

The Shannon-McMillan-Breiman theorem and the ergodic theorem implies the convergence ν-a.e. of the term into square bracket to the value

\frac{θ}{2} + P_{ϕ}(Σ) + h_{ν} + ∫ ϕdν,

which cannot be positive according to Fatou’s lemma. □

Proof of Proposition 1. Take a Markov partition of sufficiently small diameter and denote by π: Σ → M the semi-conjugacy. We know that the diameter of the image by π of a (−n, n)-cylinder goes uniformly to zero at an exponential rate. Thus, setting A = π^{−1}V, since ∂A ⊂ π^{−1}∂V, we get that the (−n, n)-cylindrical neighborhood of ∂A has a measure exponentially small, therefore Proposition 7 applies. □

5.2. Coincidence for positive values of α. Lemma 3.1 characterizes Ψ_{B_{m}} and Ψ_{C_{m}}: as soon as Ψ_{B_{m}}(α) is defined, Ψ_{B_{m}}(α) is the unique real number t = t(α, m) such that the topological pressure associated to the potential ϕ − t1_{B_{m}}, P_{ϕ−t1_{Bm}}, equals P_{ϕ}(Σ) − α.

Similarly, Ψ_{C_{m}}(α) is the unique real number t = t(α, m) such that the topological pressure associated to the potential ϕ − t1_{C_{m}}, P_{ϕ−t1_{Cm}}, equals P_{ϕ}(Σ) − α.

Let us pick some α > 0. We denote by m_{B_{m},α} the measure m_{S} obtained when we have R = B_{m} and S = P_{ϕ}(Σ) − α in subsection 3.1. This measure is the unique equilibrium state associated to the potential ϕ − Ψ_{B_{m}}(α)1_{B_{m}}. The measure weights B_{m}, hence C_{m} and we can take the induced measure on C_{m}. Therefore we have (omitting the subscribe m for convenience)

h_{m_{B,α}}(f) + ∫ ϕ − Ψ_{B}(α) dm_{B,α} = P_{ϕ}(Σ) − α
h_{m_{B,α}}(f) + ∫ ϕ − (P_{ϕ}(Σ) − α) dm_{B,α} = m_{B,α}(B)Ψ_{B}(α)
m_{B,α}(C) (h_{μ_{B,α,C}}(g_{C}) + ∫ S_{r_{C}} (ϕ − (P_{ϕ}(Σ) − α))dμ_{B,α,C}) = m_{B,α}(B)Ψ_{B}(α),

where μ_{B,α,C} is the conditional measure m_{B,α}_{|C} and g_{C} is the first return map on C. This measure has a pressure in C lower than Ψ_{C}(α); we thus get

(10)  \frac{m_{B_{m},α}(B_{m})}{m_{B_{m},α}(C_{m})}Ψ_{B_{m}}(α) ≤ Ψ_{C_{m}}(α).

Recall that for positive α, 0 < Ψ_{C_{m}}(α) ≤ Ψ_{B_{m}}(α) and are upper bounded (uniformly in every compact set in [0, α_{in}[).

Proposition 8. There exists some \overline{α} > 0 such that for every α ∈ (0, \overline{α}),

lim_{m→+∞} \frac{m_{B_{m},α}(D_{m})}{m_{B_{m},α}(C_{m})} = 0.

In particular, Ψ_{in} = Ψ_{out} on this interval.

The proof of the proposition is an immediate consequence of these two lemmas and Inequality (10).

Lemma 5.1. For any α ∈ (0, α_{in}) we have lim inf_{m→+∞} m_{B_{m},α}(C_{m}) > 0

Proof. Let us denote by μ_{m} the measure m_{B_{m},α}, and pick any accumulation point ν of (μ_{m}) such that μ_{m}(B_{m}) converges (up to the correct subsequence) to L := lim inf_{m→+∞} μ_{m}(B_{m}). Let us show that L > 0 whenever α < α_{in}.

Since μ_{m} is an equilibrium state we have

(11)  h_{μ_{m}} + ∫ ϕ dμ_{m} − Ψ_{B_{m}}(α)μ_{m}(B_{m}) = P_{ϕ}(Σ) − α.

By semi-continuity for the metric entropy and the continuity of ϕ we obtain

(12)  P_{ν}(ϕ) = h_{ν} + ∫ ϕdν ≥ P_{ϕ}(Σ) − α + Ψ_{in}(α)L.

If ν(B_{j}) = 0 then this yields that ν is a σ-invariant measure for the dotted system Σ_{B_{j}}. Hence, its ϕ-pressure must be smaller than P_{ϕ}(Σ_{B_{j}}), which is by definition P_{ϕ}(Σ) − α(B_{j}). If this holds for every j then P_{ν}(ϕ) ≤ P_{ϕ}(Σ) − α_{in} (remember that (α(B_{j})) converges to α_{in}). On the other hand by (12) we had P_{ν}(ϕ) ≥ P_{ϕ}(Σ) − α, and α < α_{in}. This yields a contradiction. Therefore ν(B_{j}) > 0 for some j. Additionally, whenever m ≥ j we get μ_{m}(B_{m}) ≥ μ_{m}(B_{j}), and the later converges to ν(B_{j}) by continuity of 1_{B_{j}}. This achieves the proof of the lemma since C_{m} ⊃ B_{m} for any m. □

Lemma 5.2. Let us set \overline{α} := min(P_{ϕ}(Σ) − P̃_{ϕ}(∂A), α_{in}) > 0. For every α ∈ (0, \overline{α}) we have lim_{m→+∞} m_{B_{m},α}(D_{m}) = 0.

Proof. Let us fix some α ∈ (0, \overline{α}). Let us pick any accumulation point ν for the sequence of measures μ_{m} (we keep the notation of the preceding lemma). We claim that ν(∂A) = 0.

Assume for a contradiction that ν(∂A) > 0. Then let H = ∪_{n∈ℤ}f^{−n}∂A be the invariant hull of ∂A. Let ν_{0} and ν_{1} be the conditional measures of ν on H and Σ\H. These two invariant probabilities are such that ν = pν_{0} +qν_{1} for some p > 0. Observe that by definition, any ergodic component of ν_{0} gives mass to H. Therefore even if ν_{0} is not ergodic, since the entropy is affine we still get that

(13)  h_{ν_{0}} + ∫ ϕ dν_{0} ≤ P̃_{ϕ}(∂A) = P_{ϕ}(Σ) − \overline{α}.

Copying the equality (11), we get for every integers m ≥ j that

h_{μ_{m}} + ∫ ϕdμ_{m} − Ψ_{B_{m}}(α)μ_{m}(B_{j}) ≥ P_{ϕ}(Σ) − α.

Thus letting m → ∞ gives, since the entropy is semi-continuous and affine, (14)

p (h_{ν_{0}} + ∫ ϕ dν_{0} − Ψ_{in}(α)ν_{0}(B_{j}))+q (h_{ν_{1}} + ∫ ϕ dν_{1} − Ψ_{in}(α)ν_{1}(B_{j})) ≥ P_{ϕ}(Σ)−α,

Hence (13) and (14) yield that for every j

h_{ν_{1}} + ∫ ϕ dν_{1} − Ψ_{in}(α)ν_{1}(B_{j}) ≥ P_{ϕ}(Σ) − α + \frac{p}{q}(\overline{α} − α).

We now choose j large enough such that

h_{ν_{1}} + ∫ ϕ dν_{1} − Ψ_{B_{j}}(α)ν_{1}(B_{j}) > P_{ϕ}(Σ) − α

holds. This is a contradiction because the measure ν_{1} would have a ϕ − Ψ_{B_{j}}1_{B_{j}}- pressure strictly larger than the associated equilibrium state. Thus we have ν(∂A) = 0.

To finish the proof let us fix some ε > 0 and consider any j such that ν(D_{j}) < ε. Such an integer j exists by outer regularity of the measure ν and because ∂A = ⋂ ↓D_{n}. Note that 1_{D_{j}} is continuous. Now, for any m ≥ j we have D_{m} ⊂ D_{j}, and then we get

0 ≤ lim sup_{m} μ_{m}(D_{m}) ≤ ν(D_{j}) < ε.

This holds for every positive ε, which proves the lemma. □

Remark 3. We remark that under the assumption in statement 1 of the theorem, we always have ν(A) = 0 for the measure ν constructed in Lemma 5.2, therefore \overline{α} = α_{in}.

5.3. Coincidence for negative values of α. We remark that the measure m_{C,α} is a Gibbs measure with full topological support, thus it gives weight to B. Therefore we can copy the case α positive and induce on B (instead of C); we get similarly

(15)  \frac{m_{C_{m},α}(C_{m})}{m_{C_{m},α}(B_{m})}Ψ_{C_{m}}(α) ≤ Ψ_{B_{m}}(α).

Proposition 9. There exists some real \underline{α} < 0 such that for every α ∈ (\underline{α}, 0),

lim_{m→+∞} \frac{m_{C_{m},α}(D_{m})}{m_{C_{m},α}(C_{m})} = 0.

In particular, Ψ_{in} = Ψ_{out} on this interval.

The proof of the proposition is an immediate consequence of these two lemmas and Inequality (15).

Lemma 5.3. For any negative α we have lim inf_{m→+∞} m_{C_{m},α}(C_{m}) > 0.

Proof. Let us denote by μ_{m} the measure m_{C_{m},α}, and pick any accumulation point ν of (μ_{m}) such that μ_{m}(C_{m}) converges (up to the correct subsequence) to L :=

lim inf_{m→+∞} μ_{m}(C_{m}).

Since μ_{m} is an equilibrium state we have

(16)  h_{μ_{m}} + ∫ ϕ dμ_{m} − Ψ_{C_{m}}(α)μ_{m}(C_{m}) = P_{ϕ}(Σ) − α.

By semi-continuity for the metric entropy and the continuity of ϕ we obtain

(17)  P_{ν}(ϕ) = h_{ν} + ∫ ϕdν ≥ P_{ϕ}(Σ) − α + Ψ_{out}(α)L.

Therefore L ≠ 0 otherwise the right hand side would be larger than the topological pressure of ϕ. □

Lemma 5.4. There exists \underline{α} < 0 such that for any α ∈ (\underline{α}, 0) we have lim_{m→∞} m_{C_{m},α}(D_{m}) = 0.

Proof. We keep the notation of the preceding lemma. Let ν be an accumulation point of (μ_{m}). We first show that ν(∂A) = 0. By equality (16) we get that for any integers m ≥ j, since C_{j} ⊃ C_{m} and now Ψ_{C_{m}}(α) < 0, we have

h_{μ_{m}} + ∫ ϕdμ_{m} − Ψ_{C_{m}}(α)μ_{m}(C_{j}) ≥ P_{ϕ}(Σ) − α.

Letting m → ∞ gives, since the entropy is semi-continuous and 1_{C_{j}} is continuous, that

h_{ν} + ∫ ϕdν − Ψ_{out}(α)ν(C_{j}) ≥ P_{ϕ}(Σ) − α.

Assume for a contradiction that ν(∂A) > 0 and decompose ν = pν_{0} + qν_{1} as in the case α positive. Let δ > 0. By definition of Ψ_{out}, for any j sufficiently large we have −Ψ_{C_{j}}(α)ν_{1}(C_{j}) + δ ≥ −Ψ_{out}(α)ν_{1}(C_{j}). Since the entropy is affine, we get

p (h_{ν_{0}} + ∫ ϕdν_{0} − Ψ_{out}(α)ν_{0}(C_{j}))+q (h_{ν_{1}} + ∫ ϕdν_{1} − Ψ_{C_{j}}(α)ν_{1}(C_{j}) + δ) ≥ P_{ϕ}(Σ)−α.

This together with (13) gives

q (h_{ν_{1}} + ∫ ϕdν_{1} − Ψ_{C_{j}}(α)ν_{1}(C_{j}) + δ) ≥ P_{ϕ}(Σ)−α−p (P_{ϕ}(Σ) − \overline{α} − Ψ_{out}(α)ν_{0}(C_{j})) .

Since the pressure P_{ν_{1}}(ϕ − Ψ_{C_{j}}(α)1_{C_{j}}) ≤ P_{ϕ}(Σ) − α this implies that

q(P_{ϕ}(Σ) − α + δ) ≥ P_{ϕ}(Σ) − α − p (P_{ϕ}(Σ) − \overline{α} − Ψ_{out}(α)ν_{0}(C_{j})) .

By outer regularity of the measure ν_{0} we have ν_{0}(C_{j}) → ν_{0}(\overline{A}) ≤ max(\overline{A}) as j → ∞. Since δ is arbitrary this gives p(−\overline{α}−Ψ_{out}(α)+α) ≥ 0, which is contradictory if p > 0 and α is small enough, since the function α ↦ α − Ψ_{out}(α) max(\overline{A}) is continuous and vanishes for α = 0. Thus there exists \underline{α} < 0 such that if α ∈]\underline{α}, 0[ we have ν(∂A) = 0.

The conclusion of the lemma follows as in the positive case. □

Remark 4. We remark that under the assumption in statement 1 of the theorem, we always have ν(A) = 0 for the measure ν constructed in Lemma 5.4, therefore \underline{α} = −∞.

6. A dynamical proof of the coincidence of inner and outer approximation in the case of totally negligible boundary

In this section we give an alternative and somewhat more direct proof of the statement 1 in our theorem. By Proposition 5 and Proposition 6 it suffices to show the equality Ψ_{in} = Ψ_{out} on the interval (−∞, 0) for the set A and its complement A^{c}. The hypotheses on the boundary is completely symmetric if we replace A by A^{c}, so it is sufficient to prove the equality on the interval (−∞, 0) for the set A only. However, we also prove that the equality holds some interval (−∞, α_{in}) for some α_{in} > 0. This in turn not only implies that the rate function Φ_{A} exists on the whole interval [0, +∞) (except at discontinuity points), but also shows that the formula (1) is satisfied on some interval [0, \overline{u}) for some \overline{u} > \frac{1}{μ_{ϕ}(A)}.

6.1. Infinite rate function for return times near the boundary. Recall that D_{m} = C_{m} \ B_{m} is the m-cylindrical neighborhood of the boundary ∂A. For convenience, and for general computations, we remove the subscript ”m” and just write D. Our aim is to show that, the probability that the successive return times into D_{m} are small, is extremely small. We first prove a key lemma.

Lemma 6.1. With the assumption on ∂A, lim_{m→+∞} max(D_{m}) = 0.

Proof. Since D_{m} is decreasing the limit ρ := lim_{m→+∞} max(D_{m}) exists. For any m there exists some probability μ_{m} such that

μ_{m}(D_{m}) ≥ max(D_{m}) − \frac{1}{m}.

Let us pick any accumulation point μ for the sequence of probabilities (μ_{m}). Recall that the map 1_{D_{m}} is continuous. Let us pick some integer m. For simplicity we write converging sequences instead of converging subsequences.

μ(D_{m}) = lim_{n→∞} μ_{n}(D_{m}) ≥ lim inf_{n→∞} μ_{n}(D_{n}) ≥ lim_{n→∞} max(D_{n}) − \frac{1}{n} = ρ.

By outer regularity of the measure μ this yields that ρ ≤ lim μ(D_{m}) = μ(∂A) = 0. □

Proposition 10. For every v > 0, there exists some M = M(v) such that for every m ≥ M, Φ_{D_{m}}(v) = −∞.

Proof. Let v > 0. By Lemma 6.1 we always can consider m large enough such that

\frac{1}{μ_{ϕ}(D)} > v.

Note that D is a union of (−m, m)-cylinders. We thus can use the large deviations principle for (r_{D}^{k}) (see Remark 1) which gives

(18)  Φ_{D}(v) = lim_{n→∞} \frac{1}{n} log μ_{ϕ} \{\frac{r_{D}^{n}}{n} ≤ v\} = inf_{α̃<α′} \{−vα̃ + Ψ_{D}(α̃)\} ,

where α′ = α(D) > 0 (it thus depends on m).

We emphasize that the slope of α̃ ↦ Ψ_{D}(α̃) as α̃ goes to −∞ is \frac{1}{max(D)}. Lemma 6.1 yields the existence of some M = M(v) such that for every m ≥ M,

[figure: α̃; Ψ_{D}(α̃); α̃v; Φ_{D}(v)]

Figure 2. The rate function Φ_{D}(v) is the maximal distance between α̃v and Ψ_{D}(α̃) on the negative axis.

\frac{1}{max(D_{m})} < v. This implies by (18) that Φ_{D_{m}}(v) ≤ lim_{α̃→−∞} α̃ (v − \frac{1}{max(D)}) = −∞ (See Figure 2). □

6.2. Coincidence for positive values of α. Fix some α ∈ (0, α_{in}) and δ such that α + δ < α_{in}. For sufficiently large m, all the Ψ_{B_{m}} are defined on [0, α + δ] and equicontinuous. then choose a uniform τ in Proposition 3 such that the mass concentration holds, namely

(19)  Ψ_{B_{m}}(α) = lim \frac{1}{n} log ∫_{r_{Bm}^{n} ≤nτ} e^{αr_{Bm}^{n}} dμ_{ϕ}

for all sufficiently large m.

Let us pick some fixed positive ε. We have

(20) ∫ _{r_{B}^{n}≤nτ} e^{αr_{B}^{n}} dμ_{ϕ} ≤ ∫_{r_{B}^{n}≤r_{C}^{n(1+ε)}} e^{αr_{C}^{n(1+ε)}} dμ_{ϕ} + ∫_{r_{C}^{n(1+ε)}<r_{B}^{n}≤nτ} e^{αr_{B}^{n}} dμ_{ϕ}.

The first term in the right hand side of this equation is simply bounded by

(21)  ∫ e^{αr_{C}^{n(1+ε)}} dμ_{ϕ} ≤ e^{n(1+2ε)Ψ_{C}(α)}

provided n is sufficiently large.

We turn to the second term. The condition r_{C}^{n(1+ε)} < r_{B}^{n} ≤ nτ implies that r_{D}^{nε} ≤ nτ, hence we get

(22)  ∫_{r_{C}^{n(1+ε)}<r_{B}^{n}≤nτ} e^{αr_{B}^{n}} dμ_{ϕ} ≤ e^{αnτ} μ_{ϕ} (r_{D}^{nε} ≤ nτ) = e^{αnτ} μ_{ϕ}(r_{D}^{nε} ≤ (nε)\frac{τ}{ε}).

By Proposition 10, if we consider m ≥ M(\frac{τ}{ε}) for some fixed ε, for n large enough we get

μ_{ϕ}(r_{D}^{nε} ≤ nε\frac{τ}{ε}) ≤ e^{−2ατn},

Therefore, (22) gives for n sufficiently large

∫_{r_{C}^{n(1+ε)}<r_{B}^{n}≤nτ} e^{αr_{B}^{n}} dμ_{ϕ} ≤ 1.

Recall that Ψ_{C}(α) ≥ 0 for α ≥ 0. Then, (20) together with (21) and (19) yield that

Ψ_{B}(α) ≤ (1 + 2ε)Ψ_{C}(α)

It follows from Proposition 4 that

(23)  Ψ_{in}(α) ≤ (1 + 2ε)Ψ_{out}(α).

Letting ε go to 0 we get that Ψ_{in}(α) = Ψ_{out}(α).

6.3. Coincidence for negative values of α. We now do the proof for a fixed α < 0. Here again we omit the subscript “m” when it is not necessary. We also pick some positive ε. Then, we have:

∫ e^{αr_{B}^{n}} dμ_{ϕ} ≥ ∫_{r_{B}^{n}≤r_{C}^{n(1+ε)}} e^{αr_{C}^{n(1+ε)}} dμ_{ϕ}
(24)  ≥ ∫ e^{αr_{C}^{n(1+ε)}} dμ_{ϕ} − ∫_{r_{B}^{n}>r_{C}^{n(1+ε)}} e^{αr_{C}^{n(1+ε)}} dμ_{ϕ}.

Let us pick some positive real τ̃ which will be chosen latter. We have

∫_{r_{B}^{n}>r_{C}^{n(1+ε)}} e^{αr_{C}^{n(1+ε)}} dμ_{ϕ} ≤ μ_{ϕ} (r_{B}^{n} > r_{C}^{n(1+ε)} > n(1 + ε)τ̃) +
+μ_{ϕ} (r_{B}^{n} > r_{C}^{n(1+ε)} ∩ n(1 + ε)τ̃ ≥ r_{C}^{n(1+ε)})
≤ μ_{ϕ} (r_{C}^{n(1+ε)} > n(1 + ε)τ̃) + μ_{ϕ} (r_{D}^{nε} ≤ n(1 + ε)τ̃) .

(25)

The large deviations principle for r_{C}^{k} means

(26)  Φ_{C}(τ̃) := lim_{n→∞} \frac{1}{n} log μ_{ϕ} \{\frac{r_{C}^{n}}{n} ≥ τ̃\} = inf_{α̃<α′} \{−τ̃α̃ + Ψ_{C}(α̃)\}

for some α′ > α_{in}. Fix some j and some α̃ ∈ (0, α(B_{j})). Choose then τ̃ such that

−τ̃α̃ + Ψ_{B_{j}}(α̃) < 2Ψ_{B_{j}}(α) < 0.

Recall that on ℝ_{+} all the Ψ_{C} are lower than all the Ψ_{B}, and the converse holds on ℝ_{−}. Therefore we get for every m that

(27)  − τ̃α̃ + Ψ_{C_{m}}(α̃) < 2Ψ_{C_{m}}(α) < 0.

For n large enough, (26) and (27) yield

(28)  μ_{ϕ} (r_{C}^{n(1+ε)} > n(1 + ε)τ̃) ≤ e^{n(1+ε)(Φ_{C}(τ̃)+ε)} ≤ e^{n(1+ε)(2Ψ_{C}(α)+ε)}.

Following Proposition 10 we get

(29)  μ_{ϕ} (r_{D}^{nε} ≤ n(1 + ε)τ̃) ≤ e^{2n(1+ε)Ψ_{C}(α)}

for every large enough m and for every large enough n. Therefore (24), (28), and (29) yield for every large enough m and for every large enough n:

∫ e^{αr_{Bm}^{n}} dμ_{ϕ} ≥ e^{n(1+ε)(Ψ_{Cm} (α)−ε)} − e^{2n(1+ε)Ψ_{Cm} (α)} − e^{n(1+ε)(2Ψ_{Cm} (α)+ε)},

For fixed m, letting n go to +∞ and using Proposition 4 with α < 0 we get for every ε > 0

(30)  Ψ_{in}(α) ≥ (1 + ε)(Ψ_{out}(α) − ε).

When ε goes to 0, we get that Ψ_{in}(α) = Ψ_{out}(α).

References

1. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture notes in Math 470, Springer-Verlag, 1975.

2. J.-R. Chazottes, R. Leplaideur, Fluctuations of the nth return time for Axiom A diffeomorphisms, Disc. Cont. Dyn. Sys. 13-2 (2005) 399–411.

3. G. Keller, Equilibrium states in ergodic theory, London Mathematical Society Student Texts 42, Cambridge University Press, 1998.

4. R. Leplaideur, Local product structure for equilibrium states, Trans. Amer. Math. Soc. 352-4 (2000) 811–827.

5. A. Dembo, O. Zeitouni, Large deviations techniques & applications, Applic. Math. 38, Springer, 1998.

6. Ya. Pesin, Dimension theory in dynamical systems: contemporary views and applications, Chicago Lectures in Mathematics, Chicago University Press, 1997.

7. Ya. Pesin, B. Pitskel, Topological pressure and the variational principle for non-compact sets, Funct. Anal. Appl. 18-4 (1997) 50–63.

8. D. Ruelle, Thermodynamic formalism, Encyclopedia of mathematics and its applications, vol. 5, Addison-Wesley, 1978.

  E-mail address: renaud.leplaideur@univ-brest.fr
  E-mail address: benoit.saussol@univ-brest.fr