Lyapunov minimizing measures for expanding maps of the circle

G. Contreras ^{∗} A.O. Lopes ^{†} Ph. Thieullen ^{‡}

January 27, 2000

Abstract

We consider the set of maps f ∈ F_{α+} = ∪_{β>α}C^{1+β} of the circle which are coverings of degree D, expanding , min_{x∈S^{1}} f′(x) > 1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f-invariant probability measure μ minimizing ∫ ln f′ dμ over all f-invariant probability measures. We show there exists a set G_{+} ⊂ F_{α+}, open and dense in the C^{1+α}- topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that every f not in G_{+} is a limit in the C^{1+α}-topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy.

We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian Systems, the analog equation was introduced by R. Mañé and extended to a subcohomological equation to all the configuration space by A. Fathi.

We will also present some results for measures maximizing (or minimizing) a general potential A.

∗CIMAT, P.O. Box 402, 36000 Guanajuato, Gto., México †Instituto de Matemática, UFRGS, Porto Alegre 91501-970, Brasil. Partially supported by PRONEX-CNPq - Sistemas Dinâmicos.

‡Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France. Partially supported by CNRS URA 1169

1 Introduction

We consider the space F_{α} of C^{1+α} maps of the circle f : S^{1} → S^{1}, (where α < 1) which are covering maps of degree D, orientation-preserving and expanding:

λ(f) = min_{x∈S^{1}} f′(x) > 1.

We denote by Höl_{α}(A), the α-Hölder constant of a function A : S^{1} → ℝ:

Höl_{α}(A) = sup_{0<d(x,y)< \frac{1}{2}}\{\frac{|A(x) − A(y)|}{d(x, y)^{α}}\}

by ∥A∥_{0}, the uniform norm and by ∥A∥_{α}, the α-Hölder norm of A:

∥A∥_{α} = Höl_{α}(A) + ∥A∥_{0}.

We note C^{α}, the set of α-Hölder functions and C^{α+} the set ∪_{β>α}C^{β} equipped with the C^{α} topology. The set of C^{1+α} maps of the circle is endowed with the distance

d(f, g) = ∥\overline{f} − \overline{g}∥_{0} + ∥f′ − g′∥_{α}

where \overline{f}, \overline{g} denote the covering maps of f, g to ℝ fixing 0. The space C^{1+α} becomes a complete space and F_{α} an open set of C^{1+α}. We denote by F_{α+} the set ∪_{β>α}F_{β} equipped with the C^{1+α}-topology.

Definition. 1 Let K(f) be the set of f-invariant probability measures,

J(f) = inf{exp(∫ ln f′dν) | ν ∈ K(f)}
M(f) = {μ ∈ K(f) | ∫ ln f′ dμ = ln J(f)}.

A measure in M(f) is called Lyapunov minimizing measure.

The purpose of this paper is to show the following theorem. We recall first that strictly ergodic means minimal and uniquely ergodic.

Theorem. 2 Let α < 1.

(i) The set G_{+} of maps f in F_{α+} having a unique Lyapunov minimizing measure supported on a periodic orbit and satisfies the property of continuously varying support is open and dense in F_{α+}.

(ii) Any map f in F_{α+} \ G_{+} is a limit in the C^{1+α}-topology of maps (f_{n}) in F_{α+} admitting a unique Lyapunov minimizing measure μ_{n} such that f_{n} restricted to supp(μ_{n}) is strictly ergodic and has positive topological entropy.

The property of continuously varying support has first been introduced by Mañé in the Lagrangian setting. It can be formulated in our setting in the following way:

Definition. 3 Suppose f ∈ F_{α} admits a unique Lyapunov minimizing measure μ. We say that f satisfies the property of continuously varying support, if for any sequence (f_{n}) of F_{α} converging to f in the C^{1+α}-topology, for any Lyapunov minimizing measures μ_{n} for f_{n}, the sequence (μ_{n}) converges to μ in the weak topology and the sequence of compact sets (supp(μ_{n})) converges to supp(μ) in the Hausdorff topology.

The problem we consider here is in some sense analogous (though we do not consider the homological position) to the problems considered in the Aubry-Mather theory (see [18], [6]) for Lagrangian flows. A recent result of Mañé (see [18] and also [6], [5]) on Lagrangian flows shows that generically on the Lagrangian there is a unique measure minimizing action. The main difference of these results to our setting is that the minimization in our theorem is among invariant probabilities and in in Mañé’s Theorem, the minimization can be done among a fixed set of probabilities (not necessarilly invariant) independent of the Lagrangian L. Therefore, in the setting considered here, if we change the map, the set of invariant probabilities will also change and this is one of the main differences from [18] to our proof. The above theorem is the analogous result for the setting of expanding maps of a conjecture for Lagrangians proposed by Mañé in [18] , [19]. E. Ott also proposed a related conjecture (see [12]). In the Lagrangian case, the conjecture whether, generically on the Lagrangian, the unique maximizing measure is supported on a periodic orbit is still open. Lyapunov minimizing measures have been previously considered by [16], [22], [12], [13], [14] and [15].

In our setting, the set M(f) may not be reduced to a single probability and the more pathological case is when ln f′ is cohomologous to a constant J, that is, when there exists a positive function h : S^{1} → ℝ such that f′ = J(h ∘ f/h). In that case, J(f) = J and any invariant measure is minimizing. Nevertheless, ln f′ always satisfies a sub-cohomological equation which is one of the key ingredients of the proof of theorem 2.

Theorem. 4 For any f ∈ F_{α}, there exists a α-Hölder function h : S^{1} →]0, 1] such that f′ ≥ J(f)(h ∘ f/h) everywhere on S^{1} and f′ = J(f)(h ∘ f/h) on the support of any Lyapunov minimizing measures.

The proof of theorems 2 depends actually on similar statements where f is fixed and A = − log f′ varies among all Hölder functions with zero-pressure. We therefore develop in section 2 a theory for maximizing a general potential A.

Definition. 5 For any C^{1}-expanding map f and any Hölder function defined on S^{1}, A : S^{1} → ℝ, we call

m(A, f) = sup{∫ Adν | ν ∈ K(f)}
M(A, f) = {μ ∈ K(f) | ∫ Adμ = m(A, f)}.

Any measure in M(A, f) will be called a maximizing measure for (A, f).

Theorem. 6 Let f be a C^{1}-expanding map and α > 0. Then the set of α-Hölder functions A admiting a unique maximizing measure for (A, f) is generic in C^{α}. For such functions A, the map f is strictly ergodic on the support of its unique maximizing measure.

If A is C^{0} and has a unique maximizing measure μ, if (A_{n}) converges to A in the C^{0}-topology and μ_{n} is choosen in M(A_{n}, f) for all n, then (μ_{n}) converges to μ in the weak topology (see step one in the proof of Proposition 16). In the following definition we ask a stronger property:

Definition. 7 Let f be a C^{1}-expanding map and A be a α-Hölder function admitting a unique maximizing measure μ. We say A satisfies the property of continuously varying support, if for any sequence (A_{n}) of α-Hölder functions converging to A in the α-Hölder topology and for any measure μ_{n} choosen in M(A_{n}, f), the sequence of measures (μ_{n}) converges weakly to μ and the sequence of its compact supports (supp μ_{n}) converges to supp μ in the Hausdorff topology.

Theorem. 8 Let f be a C^{1}-expanding map.

(i) the set of A ∈ C^{α} having a unique maximizing measure μ_{A} satisfying the property of continuously varying support is dense in C^{α}.

(ii) The set G of A ∈ C^{α} having a unique maximizing measure supported on a periodic orbit and satisfying the property of continuously varying support is open in C^{α} and G_{+} = G ∩ C^{α+} is open and dense in C^{α+}.

(iii) There is a dense set in C^{α+} \ G_{+} of functions A having a unique maximizing measure which has in addition positive entropy, is supported on a strictly ergodic compac set and satisfies the property of continuously varying support.

As before, the main tool to prove Theorem 8 is the following sub-cohomological equation.

Theorem. 9 Let f be a C^{1}-expanding map and A ∈ C^{α}. Then there exists V ∈ C^{α} such that A ≤ V ∘ f − V + m(A, f). In particular A is cohomologous to m(A, f) on the support of any maximizing measure μ ∈ M(A, f), that is, A = V ∘ f − V + m(A, f) on the support of μ.

The function V should be called sub-action in analogy with Lagrangian Mechanics, KAM Theory and with [8], because it corresponds, for the discrete time version, to a sub-solution of Hamilton-Jacobi equation:

S(γ(1)) − S(γ(0)) ≤ ∫_{0}^{1} L dt − E,

where L is the Lagrangian restricted to γ(t), t ∈ (0, 1) and E is the energy. The function A plays the role of the Lagrangian, m(A, f) plays the role of energy, f plays the role of the flow and the inequality is inverted because we are maximizing and not minimizing.

The strategy of the proof of the above results is the following: in section 2 we prove theorems 6, 8 and 9 for a fixed expanding maps f ∈ F_{0}(λ, Λ). We then transfer in section 3 the above properties to prove theorems 2, 4 for varying functions f. In that case, the potential A is linked to the the function by the formula : A = − log f′ and Thermodynamic Formalism is used. Analogous results for the one-sided shift in D-symbols are also true.

2 Maximizing measures for Hölder potentials

We begin by proving that, generically, there exists a unique maximizing measure. This comes mainly from the fact that, for a compact convex set in ℝ^{N}, among the set of hyperplanes which support the convex set, the set of those hyperplanes having an intersection reduced to a single point is generic. Nontheless the proof has to be carried in infinite dimension and requires more details.

We first recall some definitions. We say that a point p is an extremal point of a compact convex set C of ℝ^{n} if p is not the mid point of a segment totally included in C. We say that p is strictly extremal if there exists a linear form which attains its maximum at the point p only. A classical result (see [21]) states that C is equal to the closed convex hull of its strictly extremal points. Theorem 6 is a direct consequence of the one where K = K(f) and H = C^{α}.

Proposition. 10 Let K be a compact convex subset of the set of probability measures on S^{1} and (H, ∥.∥_{H}) be a dense Banach space in C^{0}(S^{1}, ℝ) which embeds continuously in C^{0}(S^{1}, ℝ). Then there exists a residual set R in H (for the ∥.∥_{H})-topology) such that, for all A ∈ R, if

M(A) =^{def} {μ ∈ K | ∫ A dμ = m(A)}andm(A) =^{def} max{∫ A dμ | μ ∈ K}

then M(A) contains a unique measure.

Proof. Let {H_{n}}_{n>1} be a dense subset of the unit ball of H. Since H is dense

d(μ, μ′) =^{def} ∑_{n≥1} \frac{1}{2^{n}} | ∫ H_{n} dμ − ∫ H_{n} dμ′|

defines a metric on K compatible with the weak topology. Let us call

R_{ϵ} =^{def} {A ∈ C^{0}(S^{1}, ℝ) s.t. diam M(A) < ϵ}.

We claim that R_{ϵ} is open in C^{0}(S^{1}, ℝ) and R_{ϵ} ∩ H is dense in H for the ∥.∥_{H}-topology. The desired residual set will be R = ∩_{ϵ}R_{ϵ} ∩ H.

We show by contradiction that R_{ϵ} is open. If not, one can find B_{n} in C^{0}(S^{1}, ℝ) and (μ_{n}, ν_{n}) in M(A + B_{n}) such that ∥B_{n}∥_{0} converges to 0 and d(μ_{n}, ν_{n}) ≥ ϵ for all n. We may assume by taking a subsequence that (μ_{n}) and (ν_{n}) converge to μ̄ and ν̄. Let us prove that μ̄ ∈ M(A): indeed for every μ ∈ K

∫ (A + B_{n})dμ ≤ ∫ (A + B_{n})dμ_{n} ≤ ∫ A dμ_{n} + ∥B_{n}∥_{0}

and ∫ A dμ ≤ ∫ A dμ̄ by taking the limit on n. For the same reason ν̄ belongs to M(A). We have obtain a contradiction since d(μ̄, ν̄) ≥ ϵ.

We now show that R_{ϵ} ∩ H is dense in H. Let A_{0} ∈ H and K_{0} = M(A_{0}). The continuous projections π_{n} : K → ℝ^{n}, π_{n}(μ) = (∫ H_{1}dμ, …, ∫ H_{n}dμ) sends K_{0} to a compact convex set π_{n}(K_{0}) which admits a strictly extremal point p_{n}. We first notice that diam π_{n}^{−1}(p_{n}) ≤ 2^{−n} and we choose once for all, n large enough so that 2^{−n} < ϵ. By definition of p_{n} = (p^{1}, …, p^{n}) there exists (a^{1}, …, a^{n}) ∈ ℝ^{n} such that

∑_{i=1}^{n} a^{i}p^{i} > ∑_{i=1}^{n} a^{i}q^{i} ∀ q = (q^{1}, …, q^{n}) ∈ π_{n}(K_{0}), q ≠ p.

In particular if A_{1} = ∑_{i=1}^{n} a^{i}H_{i}, m_{0}(A_{1}) = max{∫ A_{1}dμ | μ ∈ K_{0}} and

M_{0}(A_{1}) =^{def} {μ ∈ K_{0} | ∫ A_{1}dμ = m_{0}(A_{1})},

then M_{0}(A_{1}) = π_{n}^{−1}(p_{n}) has diameter less than ϵ. We show that, for small enough ζ > 0, A_{ζ} = (1 − ζ)A_{0} + ζA_{1} ∈ R_{ϵ}. More precisely we show that, for any open set U ⊃ M_{0}(A_{1}), for any ζ sufficiently small, M(A_{ζ}) ⊂ U. By contradiction, there exists a sequence μ_{n} ∈ M(A_{ζ_{n}})\U for some ζ_{n} converging to 0. We may assume that (μ_{n}) converges to μ̄ ∈ K \ U. We first show that μ̄ ∈ K_{0} = M(A_{0}): for every μ ∈ K

∫ A_{ζ_{n}}dμ ≤ ∫ A_{ζ_{n}}dμ_{n} ≤ ∫ A_{0} dμ_{n} + ζ_{n}∥A_{1} − A_{0}∥_{0}

and by taking a limit in n: ∫ A_{0}dμ ≤ ∫ A_{0}dμ̄. We then show that μ̄ belongs to M_{0}(A_{1}): for every μ ∈ K_{0}

∫ A_{ζ_{n}}dμ = (1 − ζ_{n}) ∫ A_{0} dμ + ζ_{n} ∫ A_{1} dμ ≤ (1 − ζ_{n}) ∫ A_{0} dμ_{n} + ζ_{n} ∫ A_{1} dμ_{n}.

Since ∫ A_{0} dμ_{n} ≤ ∫ A_{0} dμ, we obtain ∫ A_{1} dμ ≤ ∫ A_{1} dμ_{n} and ∫ A_{1} dμ ≤ ∫ A_{1} dμ̄. We have obtain a contradiction since μ̄ ∉ U. ■

We now prove the cocycle theorem 9. We show that any A ∈ C^{α} is subcohomologous to a constant m(A, f) with a unique minimal coboundary V_{A}. The main tool is the shadowing lemma. We say that two points x and y belong to the same inverse branch of length n if there exists a close interval I containing x and y such that f^{n} restricted to int(I) is one-to-one and onto S^{1}.

Proposition. 11 Let f be a C^{1}-expanding map and A ∈ C^{α}. Then there exists a unique minimal non-negative α-Hölder function V_{A} : S^{1} → ℝ such that

(i) A ≤ V_{A} ∘ f − V_{A} + m(A, f) on S^{1},

(ii) for any non-negative function W : S^{1} → ℝ (not necessarily Hölder) satisfying A ≤ W ∘ f − W + m(A, f) on S^{1}, we have V_{A} ≤ W on S^{1}.

Moreover Höl_{α}(V_{A}) ≤ Höl_{α}(A)/(λ^{α} − 1) and A = V_{A} ∘ f − V_{A} + m(A, f) everywhere on supp(μ) of any maximizing measure μ for (A, f).

Proof. We may assume m(A, f) = 0. We recall that S_{n}A denotes the Birkhoff sum ∑_{k=0}^{n−1} A ∘ f^{k} and that S_{n}A = 0 when n = 0. Let us define

V_{A}(x) = sup{S_{n}A(y) | n ≥ 0 and f^{n}(y) = x}.

We show that V_{A} is well defined. On the one hand, for any periodic point p of period n, S_{n}A(p) ≤ 0. On the other hand, for any n ≥ 0 and any y ∈ S^{1} there exists a periodic point p in the same inverse branch of length n as y such that f^{n}(p) = p . Moreover if N is large enough, λ^{N} > 2, for all k < n − N, d(f^{k}(y), f^{k}(p)) < \frac{1}{2} and

|S_{n}A(y) − S_{n}A(p)| ≤ \frac{Höl_{α}(A)}{λ^{α} − 1} + 2N∥A∥_{0}.

We now show that V_{A} is α-Hölder. If f^{n}(y) = x, f^{n}(y′) = x′ and y and y′ are in the same inverse branch of length n, if d(x, x′) < \frac{1}{2} then

|S_{n}A(y) − S_{n}A(y′)| ≤ \frac{Höl_{α}(A)}{λ^{α} − 1}d(x, x′)^{α}

and by taking supremum over y and n, we finally obtain

|V_{A}(x) − V_{A}(x′)| ≤ \frac{Höl_{α}(A)}{λ^{α} − 1}d(x, x′)^{α}.

By definition of V, we have V_{A} ∘ f ≥ A + V_{A} on S^{1}. If W is non-negative and satisfies W ∘ f ≥ A + W on S^{1}, then for every n ≥ 0, (x, y) such that f^{n}(y) = x, W (x) ≥ S_{n}A(y) + W (y) and therefore W (x) ≥ V_{A}(x). If μ is a maximizing measure, ∫(V_{A} ∘ f − V_{A} − A)dμ = 0 and V_{A} ∘ f − V_{A} = A on supp(μ). ■

Remark. 12

(i) If we choose in Proposition 11 the following definition of Ṽ_{A}:

Ṽ_{A}(x) = lim_{n→∞} sup_{k>n} sup_{f^{k}(y)=x}{S_{k}(A(y) − km(A, f) =},

we obtain again a sub-coboundary which satisfies in addition the functional equation considered by Th. Bousch [2] and A. Fathi [8]:

∀x ∈ S^{1}, Ṽ_{A}(x) = max_{f(y)=x}{A(y) − m(A, f) + Ṽ_{A}(y)}.

In particular, for any opposite points x, x′ (i.e. f(x) = f(x′)), one of them belongs to the set {A − m(A, f) = Ṽ_{A} ∘ f − Ṽ_{A}} which has therefore always a non empty interior.

(ii) If A is not equal to a coboundary modulo a constant then the set of α-Hölder functions W solution of A ≤ W ∘ f − W + m(A, f) is always a large set. Indeed let R = V_{A} ∘ f − V_{A} − A + m(A, f) and 1 ≥ α_{0} ≥ α_{1} ⋯ ≥ α_{n} ≥ 0, α-Hölder functions, then

W = V_{A} + α_{0}R + α_{1} ∘ fR ∘ f + ⋯ + α_{n} ∘ f^{n}R ∘ f^{n}

is such a solution.

(iii) Proposition 11 shows that, in the case A admits a unique maximizing measure μ_{A}, then f is uniquely ergodic on supp(μ_{A}). This follows from the fact that any measure ν with support contained in the support of μ satisfies ∫ Adν = m(A, f). A maximizing measure (on S^{1}) for A may not be unique: the more pathological case is when A is of the form A = m + V ∘ f − V, where any invariant measure is maximizing.

In order to prove the property of continuously varying support, we introduce a vocabulary close to what is used in the setting of Lagrangians flows.

Definition. 13 Let A be a continuous function and m = m(A, f).

(i) A function W is called a ”sub-action” for (A, f) if it satisfies the inequality A − m ≤ W ∘ f − W everywhere. We then call the set

{A − m = W ∘ f − W }

the W -action set (we could have called it the Aubry-set).

(ii) If W is a sub-action and (x, y) is a pair of points of S^{1}, we say that the points (x, y) are W -connected and we write x →^{W} y if, for every ϵ > 0, there exist z ∈ S^{1} and n ≥ 1 such that d(z, x) < ϵ, f^{n}(z) = y and

|S_{n}(A − m)(z) − (W (y) − W (x))| < ϵ.

We say that (x, y) are equivalent if x →^{W} y and y →^{W} x.

(iii) A point x ∈ S^{1} is said to be recurrent with respect to (A, f) if, for any ϵ > 0, there exist z ∈ S^{1} and n ≥ 1 such that

d(z, x) < ϵ, f^{n}(z) = x and |S_{n}(A − m)(z)| < ϵ.

We denote by Ω(A, f), the set of non-wandering points with respect to (A, f).

(iv) If W is a sub-action, a compact invariant set K, (i.e. f(K) = K), is said to be W -irreducible if any two points (x, y) of K are W -equivalent.

We first give elementary properties. The main tool is the standard shadowing lemma. We recall that, for a complete orbit \underline{x} = (x_{n})_{n∈ℤ} in S^{1} (f(x_{n}) = x_{n+1} for all n ∈ ℤ), ω(\underline{x}) denotes the compact invariant set of accumulation points of (x_{n})_{n≥0} and α(\underline{x}) the compact invariant set of accumulation points of (x_{−n})_{n≥0}. We also recall that an ϵ-pseudo orbit with M jumps from x to y is a finite sequence of points {x_{0} … x_{n}} such that x_{0} = x, x_{n} = y, d(f(x_{k}), x_{k+1}) < ϵ for all 0 ≤ k < n and the cardinality of the set of indices 0 ≤ k < n satisfying f(x_{k}) ≠ x_{k+1} is bounded by M.

Lemma. 14 Let W be a Hölder sub-action.

(i) Let (x, y) be a pair of points and M > 0 an integer. Then x →^{W} y if and only if, for every ϵ > 0 there exists a ϵ-pseudo orbit with at most M jumps from x to y such that

| ∑_{k=0}^{n−1}(A − m(A, f))(x_{k}) − (W (y) − W (x))| < ϵ.

(ii) If x →^{W} y and y →^{W} z then x →^{W} z.

(iii) Any W -irreducible compact invariant set K is included in Ω(A, f).

(iv) If \underline{x} = (x_{n})_{n∈ℤ} is a complete orbit in the W -action set, then α(\underline{x}) and ω(\underline{x}) are compact invariant W -irreducible sets and therefore belong to

Ω(A, f). For any α ∈ α(\underline{x}) and ω ∈ ω(\underline{x}), α →^{W} x →^{W} ω

The main ingredients for proving Theorem 8 are given by the following two propositions.

Proposition. 15 Let A be a Hölder function. Then :

(i) Ω(A, f) is a compact invariant set included in the W -action set of any continuous sub-action W.

(ii) For any invariant measure μ, μ ∈ M(A, f) if and only if supp(μ) ⊂ Ω(A, f). In particular (A, f) admits a unique maximizing measure if and only if Ω(A, f) is uniquely ergodic. If μ is an ergodic maximizing measure, supp(μ) is W -irreducible for any continuous sub-action W.

(iii) If Ω(A, f) is W -irreducible for some Hölder sub-action W, then Ω(A, f) contains all compact invariant set included in the W -action set.

(iv) If (A, f) admits a unique maximizing measure, then Ω(A, f) is W - irreducible for any Hölder sub-action W.

Proof. Part (i). If (x_{i})_{i≥0} is a sequence of points of Ω(A, f) converging to x and ϵ > 0, then d(x_{i}, x) < \frac{1}{2}ϵ for some i and there exist y_{i}, n > 0 such that d(y_{i}, x_{i}) < \frac{1}{2}ϵ, d(f^{n}(y_{i}), x_{i}) < \frac{1}{2}ϵ and |S_{n}(A − m(A, f))(y_{i})| < ϵ. Then d(y_{i}, x) < ϵ and d(f^{n}(y_{i}), x) < ϵ. We have proved Ω(A, f) is closed.

Let x be a point of Ω(A, f) and ϵ > 0. We choose, using the continuity of f and A, η > 0 so that f(B(x, η)) ⊂ B(f(x), ϵ) and the oscilation of A on B(x, η) is bounded by ϵ. Then there exist y andn > 0 such that d(y, x) < η, d(f^{n}(y), x) < η and |S_{n}(A − m(A, f))(y)| < ϵ. Using the definition of η, we obtain d(f(y), f(x)) < η, d(f^{n+1}(y), f(x)) < η and

|S_{n}(A − m(A, f))(f(y))| = |S_{n}(A − m(A, f))(y) − (A − A ∘ f^{n})(y)| < 2ϵ.

We have proved f(Ω(A, f)) ⊂ Ω(A, f).

If x′ ∈ Ω(A, f), (y′_{i})_{i≥0} is a sequence of points in S^{1} converging to x′, (n_{i})_{i≥0} an increasing sequence of integers such that f^{n_{i}}(y′_{i}) converges to x′ and |S_{n_{i}}(A − m(A, f))(y′_{i})| converges to 0, by taking a subsequence, we may assume f^{n_{i}−1}(y′_{i}) converges to some x satisfying f(x) = x′. For any large i, y′_{i} admits a unique pre-image y_{i} close to x and as above |S_{n_{i}}(A − m(A, f))(y_{i})| converges to 0. We have proved f(Ω(A, f)) = Ω(A, f).

To prove that Ω(A, f) is included in the W -action set, we introduce the non-negative function :

R = W ∘ f − W + m(A, f) − A.

If x ∈ Ω(A, f), (y_{i})_{i≥0} converges to x and (n_{i})_{i≥0} is choosen so that f^{n_{i}}(y_{i}) converges to x and S_{n_{i}}(A − m(A, f))(y_{i}) converges to 0, then

S_{n_{i}}(A − m(A, f))(y_{i}) + S_{n_{i}}R(y_{i}) = W ∘ f^{n_{i}}(y_{i}) − W (y_{i})

converges to 0 and therefore R(x) = 0.

Part (ii). The fact that the support of any maximizing measure μ is included in Ω(A, f) follows from Atkinson’s theorem [1] since, by definition of μ, ∫(A − m(A, f))dμ = 0. If, in addition μ is ergodic, for any two points (x, y), there exists a path {z, …, f^{n}(z)} included in the support of μ which connects as close as we want (x, y). Since that path is included in the W - action set, we just have shown that supp(μ) is W -irreducible.

Part (iii). We now assume Ω(A, f) is W -irreducible for some Hölder subaction W. Let K be a compact invariant set included in the W -action set. We pick \underline{z} = (z_{n})_{n∈ℤ}, a complet orbit in K passing through z (i.e. z_{0} = z), α ∈ α(\underline{z}) and ω ∈ ω(\underline{z}). Then α →^{W} z →^{W} ω and by irreducibility of Ω(A, f), ω →^{W} α. Therefore z →^{W} z and z ∈ Ω(A, f).

Part (iv). The proof is very similar to Part (iii). Let μ be the unique maximizing measure, x, y, points of Ω(A, f), \underline{x} = (x_{n})_{n∈ℤ}, \underline{y} = (y_{n})_{n∈ℤ} complete orbits passing through x, y respectively. We define two sequences of probability measures :

ω_{n} = \frac{1}{n} ∑_{k=0}^{n−1} δ_{x_{k}} and α_{n} = \frac{1}{n} ∑_{k=0}^{n−1} δ_{y_{−k}}.

By taking subsequences, (ω_{n′}) and (α_{n′}) converge to the same measure μ and in particular one can find ω ∈ ω(\underline{x}) ∩ supp(μ) and α ∈ α(\underline{y}) ∩ supp(μ). Since

x →^{W} ω, α →^{W} y and ω →^{W} α by ergodicity of μ, we obtain x →^{W} y for all (x, y) ∈ Ω(A, f) : Ω(A, f) is W -irreducible. ■

Proposition. 16 Let A : S^{1} → ℝ be a α-Hölder function.

(i) If Ω(A, f) is minimal then, for any sequence (A_{n})_{n≥0} of α-Hölder functions converging to A in the C^{α}-topology, the sequence of compact sets (Ω(A_{n}, f))_{n≥0} converges to Ω(A, f) in the Hausdorff topology.

(ii) If Ω(A, f) is strictly ergodic, A satisfies the property of continuously varying support.

Proof. Step one. Using only the convergence of (A_{n})_{n≥0} to A in the C^{0}- topology, we show that (m(A_{n}, f))_{n≥0} converges to m(A, f) and that, if μ_{n} is some maximizing measure for (A_{n}, f), any weak limite of (μ_{n})_{n≥0} is again a maximizing measure for (A, f). Indeed we have :

|m(A, f) − m(B, f)| ≤ ||A − B||_{0}

for all C^{0} A, B and for any sub-sequence (μ_{n′})_{n′} converging to some measure μ

∫ Adμ = lim_{n′} ∫ A_{n}dμ_{n′} = lim_{n′} m(A_{n′}, f) = m(A, f).

Step two. Let K_{n} = Ω(A_{n}, f) and K be a limit set in the Hausdorff topology of some sub-sequence (K_{n′})_{n′}. For each n, there exists a α-Hölder sub-action V_{n} for (A_{n}, f) such that K_{n} is included in the V_{n}-action set :

A_{n} − m(A_{n}, f) = V_{n} ∘ f − V_{n} on K_{n}.

We may assume that V_{n} is normalized so that V_{n} = 0 for some point in K_{n}. Since (A_{n})_{n} converges to A in the C^{α}-topology, using Proposition 11, we may also assume that the α-Hölder norm of V_{n} is uniformly bounded. By Ascoli’s Theorem, we can extract a sub-sequence (V_{n′})_{n′} which converges uniformly to a α-Hölder function V. We thus have obtained a sub-action V ∈ C^{α} for (A, f) and a compact invariant set K, (i.e. f(K) = K) included in the V - action set. Since Ω(A, f) is irreducible by transitivity, K has to be included in Ω(A, f) and therefore has to be equal to Ω(A, f) by minimality.

Step three. We assume now that Ω(A, f) is strictly ergodic and is equal to the support of the unique ergodic measure μ. Let μ_{n} be some maximizing measure for (A_{n}, f) and K_{n} = supp(μ_{n}). Since K_{n} is included in Ω(A_{n}, f), any accumulation set of (K_{n})_{n≥0} is contained in Ω(A, f) and equals Ω(A, f) by minimality. Any weak-accumulation measure has a support included in Ω(A, f) and is therefore equal to μ by unique ergodicity. ■

We are now able to prove the first part of Theorem 8

Proof of Theorem 8, part (i). If A is α-Hölder function and admits a unique maximizing measure μ, we can find ϕ ∈ C^{α}, with small ∥.∥_{α}-norm, such that ϕ = 0 on supp(μ) and ϕ > 0 everywhere else. Then m(A − ϕ, f) = m(A, f) and A−ϕ admits the same unique maximizing measure μ. Moreover V_{A} is again a sub-action for A − ϕ and

{A − ϕ − m(A − ϕ, f) = V_{A} ∘ f − V_{A}} = supp(μ).

Therefore Ω(A−ϕ, f) = supp(μ) is uniquely ergodic and according to Proposition 16, A − ϕ satisfies the property of continuously varying support.

The strategy of the proof of Theorem 8, part (ii) is to find a periodic orbit well distributed and closed to Ω(A, f). We begin by proving a theorem of approximation in the C^{α}-topology for functions having a better regulartity C^{β}, β > α.

Lemma. 17 Let 0 < α < β < 1. For any β-Hölder functions A:

Höl_{α}(A) ≤ 2Höl_{β}(A)^{\frac{α}{β}} ∥A∥_{0}^{1− \frac{α}{β}} .

In particular, if (A_{n}) is a sequence of β-hölder functions uniformly bounded in the C^{β}-topology, if (A_{n}) converges to 0 in the C^{0}-topology, then (A_{n}) converges to 0 in the C^{α}-toplogy.

Proof. The following two estimates

|A(x) − A(y)| ≤ Höl_{β}(A)d(x, y)^{β} ≤ Höl_{β}(A)d(x, y)^{β−α}d(x, y)^{α},

|A(x) − A(y)| ≤ 2∥A∥_{0} ≤ \frac{2∥A∥_{∞}}{d(x, y)^{α}} d(x, y)^{α},

shows that A is α-Hölder with

Höl_{α}(A) ≤ 2 sup_{x≠y} min(\frac{∥A∥_{0}}{d(x, y)^{α}} , Höl_{β}(A)d(x, y)^{β−α}).

The minimum is reached at the intersection of the decreasing graph t ↦ ∥A∥_{0}t^{−α} and the increasing graph t ↦ Höl_{β}(A)t^{β−α}. ■

Although it is a very standard lemma, we need a precise estimate of the shadowing constant in the proof of Lemma 19 and we give a proof.

Lemma. 18 Let f be C^{1}-expanding, λ = λ(f) and N such that λ^{N} > 2. Then for any x ∈ S^{1} such that d(x, f^{N}(x)) is less than \frac{1}{4}, there exists a periodic point p of period N such that

∀ 0 ≤ k ≤ N d(f^{k}(p), f^{k}(x)) ≤ \frac{λ^{N}}{λ^{N} − 1} \frac{d(x, f^{N}(x))}{λ^{N−k}}.

Notice that λ^{N}/(λ^{N} − 1) → 1 when N → +∞.

Proof. We call x_{1} = f^{N}(x) and x_{0} = x. We assume x_{1} ≠ x_{0} (otherwise we choose p = x_{0}) and we denote by ]a, b[ the smallest unordered interval joining a and b. Let x′_{0} be the closest preimage of x_{1}, f^{N}(x′_{0}) = x_{1}, such that f^{N} restricted to ]x′_{0}, x_{0}[ is an homeomorphism onto S^{1} \ {x_{1}} and such that ]x′_{0}, x_{0}[ is disjoint from ]x_{0}, x_{1}[ (we use the fact that λ^{N} > 2). Let x_{1} be the unique preimage of x_{0} in ]x′_{0}, x_{0}[ by the map f^{N}, then f^{N}(]x_{−1}, x_{0}[) =]x_{0}, x_{1}[ (we use the fact that f^{N} preserves the orientation). By induction we obtain a sequence (x_{−k}) of points ]x′_{0}, x_{0}[ such that (]x_{−k}, x_{−k+1}[) are pairwise disjoint and such that f^{N} maps ]x_{−k−1}, x_{−k}[ onto ]x_{−k}, x_{−k+1}[. The point p, limit of (x_{−k}) is therefore a periodic point of period N and the length of the interval ]p, x_{1}[ is bounded by

d(x_{0}, x_{1})[1 + \frac{1}{λ^{N}} + \frac{1}{λ^{2N}} + …] ≤ \frac{λ^{N}}{λ^{N} − 1}d(x_{0}, x_{1}).

In particular, this length is bouded by \frac{1}{2}, f^{N} maps ]p, x_{0}[ onto ]p, x_{1}[ and

∀ 0 ≤ k ≤ N d(k^{k}(p), f^{k}(x_{0})) ≤ \frac{1}{λ^{N−k}} d(p, x_{1}).

The two inequalities combined end the proof. ■

Lemma. 19 Let Ω be an invariant compact set, f(Ω) = Ω, without periodic orbit. Let η = \frac{1}{2}(1 − \frac{1}{λ}). Then for any ϵ > 0 sufficiently small, there exists a periodic orbit {p, f(p), …, f^{N}(p) = p} such that

d(f^{i}(p), Ω) < ϵ and d(f^{i}(p), f^{j}(p)) > ηϵ,

for all 0 ≤ i < j < N.

Proof. For each K > 0 we define

d(K) = min{d(f^{i}(x), f^{j}(x)) | x ∈ Ω, 0 ≤ i < j ≤ K}.

Since no periodic orbit belongs to Ω, d(K) > 0 and tends to 0 when K tends to ∞. We now choose K so that λ^{K} > 5 and

1 − \frac{λ^{K}}{λ^{K} − 1}(\frac{1}{λ} + \frac{1}{λ^{K}}) > η = .

Let ϵ be small enough such that ϵ < \frac{1}{3}d(K). We first exibit an almost closed orbit {x, f(x), …, f^{N}(x)} in Ω satisfying

d(x, f^{N}(x)) < ϵ and d(f^{i}(x), f^{j}(x)) ≥ ϵ

for all 0 ≤ i < j ≤ N except i = 0 and j = N. This can be achieved by choosing an orbit {y, f(y), …, f^{n}(y)} in Ω such that d(y, f^{n}(y)) < ϵ and a sub-orbit {x, f(x), …, f^{N}(x)} with the property that d(x, f^{N}(x)) < ϵ and N is minimal (x is a point in the orbit of y). Since ϵ < d(K), N has to be bigger than K. By the shadowing lemma 18 there exists a periodic point p of period N such that

d(f^{i}(p), f^{i}(x)) < \frac{λ^{N}}{λ^{N} − 1} \frac{ϵ}{λ^{N−i} =},

for all 0 ≤ i ≤ N. We now show that p is the good gandidate.

For 0 ≤ i < j ≤ N − K,

d(f^{i}(p), f^{i}(x)) < \frac{ϵ}{4}, d(f^{j}(p), f^{j}(x)) < \frac{ϵ}{4}, d(f^{i}(x), f^{j}(x)) ≥ ϵ

and we obtain d(f^{i}(p), f^{j}(p)) > \frac{1}{2}ϵ > ηϵ.

For N − K ≤ i < j ≤ N

d(f^{i}(p), f^{i}(x)) < ϵ, d(f^{j}(p), f^{j}(x)) < ϵ, d(f^{i}(x), f^{j}(x)) ≥ d(K) > 3ϵ

and we obtain in this case d(f^{i}(p), f^{j}(p)) > ϵ > ηϵ.

For 0 ≤ i < N − K < j < N, d(f^{i}(x), f^{j}(x)) ≥ ϵ and

d(f^{i}(p), f^{i}(x)) < \frac{λ^{K}}{λ^{K} − 1} \frac{ϵ}{λ^{K}} , d(f^{j}(p), f^{j}(x)) < \frac{λ^{K}}{λ^{K} − 1} \frac{ϵ}{λ},

and we obtain d(f^{i}(p), f^{j}(p)) > ϵ(1 − \frac{λ^{K}}{λ^{K}−1}(\frac{1}{λ} + \frac{1}{λ^{K}})) > ηϵ. ■

Proof of Theorem 8, part (ii). Let G be the set of functions A ∈ C^{α} having a unique maximizing measure supported on a periodic orbit and satisfying the property of continuously varying support. We first show that G is open in C^{α}. Let A_{0} ∈ G and μ_{0} = \frac{1}{n} ∑_{k=0}^{n−1} δ_{f^{k}(p)} the unique maximizing measure supported on a periodic orbit p. By the property of continuously varying support, there exist a neighborhood N of orb(p) and a neighborhood U in H_{α} of A_{0} such that for any A ∈ U and any μ ∈ M(A, f), the support of μ is included in N. By the standard shadowing lemma, μ = μ_{0} is the only invariant measure close to orb(p). Then U ∪ H_{α+} ⊂ G.

We now show that G_{+} = G ∩ C^{α+} is dense in C^{α+}. Let A_{1} ∈ C^{α+}, then A_{1} ∈ C^{β}, for some α < β < 1. We look for some A = A_{1} + ϕ, ϕ ∈ C^{β} with small ∥.∥_{α}-norm such that Ω(A, f) is reduced to a periodic orbit. We actually prove the existence of a sequence (ϕ_{n}) of C^{β}-functions having a βnorm ,uniformly bounded and converging to 0 in the uniform topology.

¿From Proposition 11 and 15 (i), we can write

A_{1} = m(A_{1}, f) + V_{1} ∘ f − V_{1} − R_{1}

where V_{1}, R_{1} ∈ C^{β}, R_{1} ≥ 0 and R_{1} = 0 on Ω(A_{1}, f).

We first construct some ϕ_{1} ∈ C^{β}, ϕ_{1} ≥ 0 with small ∥.∥_{α}-norm such that A_{1} + ϕ_{1} admits a maximizing measure supported on a periodic orbit. If Ω(A, f) already contains a periodic orbit, we choose ϕ_{1} = 0. If not, from Lemma 19, for any ϵ > 0, there exists a periodic orbit orb(p) = {p, …, f^{N}(p) = p} satisfying d(q, Ω(A_{1}, f)) < ϵ and d(q, q′) > ηϵ for any q, q′ ∈ orb(p), q ≠ q′, where η = \frac{1}{2}(1 − \frac{1}{λ}). We note

C_{1} = Höl_{β}(R_{1}).

Since R_{1} = 0 on Ω(A_{1}, f) and |R_{1}(q)| ≤ C_{1}d(q, Ω(A_{1}, f))^{β}, we obtain for all q ∈ orb(p)

ϵ > d(q, Ω(A_{1}, f)) ≥ (\frac{R_{1}(q)}{C_{1}})^{\frac{1}{β}} .

We now define for each q on the orbit a localized function ϕ_{1,q}. The functions {ϕ_{1,q}}_{q∈orb(p)} have disjoint support, they satisfy 0 ≤ ϕ_{1,q} ≤ R_{1} everywhere and ϕ_{1,q}(q) = R_{1}(q) for all q ∈ orb(p), by the following formula

ϕ_{1,q}(x) = [R_{1}(q) − D_{1}(q)d(x, q)^{β}]^{+}

where D_{1}(q) = (\frac{2}{η})^{β}C_{1}. Clearly D_{1}(q) ≥ C_{1} and by the Hölder property of R_{1}, we have ϕ_{1,q} ≤ R_{1}. The support of each ϕ_{1,q} is included in the ball of center q and radius

[\frac{R_{1}(q)}{D_{1}(q)}]^{1/β} = [\frac{R_{1}(q)}{C_{1}}]^{1/β} \frac{η}{2} < \frac{1}{2}ϵη

which shows that {ϕ_{1,q}}_{q∈orb(p)} have disjoint support.

On the one hand, the Höl_{β}-seminorm of ϕ_{1,q} is uniformly bounded, independently of ϵ,

Höl_{β}(ϕ_{1,q}) ≤ C_{1} + D_{1}(q) ≤ 2(\frac{2}{η})^{β}C_{1}

(we have used : |a^{+} − b^{+}| ≤ |a − b|, |a^{β} − b^{β}| ≤ |a − b|^{β} when β < 1.) On the other hand, the ∥.∥_{∞}-norm of ϕ_{1,q} tends to 0 when ϵ tends to 0

∥ϕ_{1,q}∥_{0} ≤ R_{1}(q) ≤ C_{1}ϵ^{β}.

We thus have obtained a non-negative function

ϕ_{1} = ∑_{q∈orb(p)} ϕ_{1,q}

satisfying ∥ϕ_{1}∥_{0} ≤ C_{1}ϵ^{β}, Höl_{β}(ϕ_{1}) ≤ 4(2/η)^{β}C_{1}, 0 ≤ ϕ_{1} ≤ R_{1} everywhere and ϕ_{1}(q) = R_{1}(q) for all points q ∈ orb(p). By using Lemma 17, we obtain some ϕ_{1} ∈ C^{β} with small ∥.∥_{α}-norm. Moreover m(A_{1} + ϕ_{1}, f) = m(A_{1}, f) and Ω(A_{1} + ϕ_{1}, f) contains both Ω(A_{1}, f) and orb(p). Indeed

A_{1} + ϕ_{1} = V_{1} ∘ f − V_{1} − R_{1} + ϕ_{1} + m(A_{1}, f).

If x ∈ Ω(A_{1}, f), ϵ > 0 and z, n > 0 have been chosen such that d(x, z) < ϵ, d(x, f^{n}(z)) < ϵ and | ∑_{k=0}^{n−1} R ∘ f^{k}(z)| < ϵ, then | ∑_{k=0}^{n−1} ϕ_{1} ∘ f^{k}(z)| < ϵ and we have shown that x ∈ Ω(A_{1} + ϕ_{1}, f).

We now choose ϕ_{2} ≥ 0, ϕ_{2} ∈ C^{β}, with small ∥.∥_{β}-norm such that ϕ_{2} = 0 on orb(p) and ϕ_{2} > 0 else where. Then A = A_{1} + ϕ_{1} − ϕ_{2} ∈ C^{β} is close to A_{0} in the C^{α}-topology and admits a unique maximizing measure μ = \frac{1}{# orb(p)} ∑_{q∈orb(p)} δ_{q}. ■

We finally prove Theorem 8, part (iii), which is actually a direct consequence of the following proposition.

Proposition. 20 Let f be a C^{1}-expanding map, G_{+} be the set defined in Theorem 8 and A ∈ C^{β}, β > α. The following are equivalent.

(i) A ∉ G_{+}.

(ii) Ω(A, f) is not reduced to the orbit of a periodic point.

(iii) A is a limit in the C^{α}-topology of C^{β}-functions (A_{n}), uniformly bounded in the C^{β}-topology, having a unique maximizing measure μ_{n} whose support is strictly egodic and has positive topological entropy.

The main difficulty in proving Proposition 20 is the implication: (ii) implies (iii). We first use Grillenberger’s theorem [10] and [11] to construct strictly ergodic compact invariant sets which have positive entropy and are close to any compact invariant sets not equal to a finite union of periodic orbits. We then show that, when Ω(A, f) is not reduced to a single periodic orbit, there exists a sub-action V such that the V -action set contains a compact invariant set which is not equal to a finite union of periodic orbits.

Lemma. 21 For any compact invariant set K not equal to a finite union of periodic orbits, for any neighborhood U of K, there exists an invariant strictly ergodic compact set L included in U and admitting a positive topological entropy.

Proof. We first assume that K is a sub-shift of B(D) (the two-sided full shift of D symbols). Let U be an open set containing K and C_{1} ∪ ⋯ ∪ C_{N} a finite union of cylinders covering B(D) \ U and disjoint from K. Let K̃ be defined by the exclusion rules:

K̃ = {x ∈ B(D) | ∀i = 1 … N ∀n ∈ ℤσ^{n}(x) ∉ C_{i}}

Then K̃ is a sub-shift of finite type containing K and included in U. Since K̃ is not equal to a finite union of periodic orbits, the topological entropy of σ | K̃ is positive. By Grillenberger, there exists L ⊂ K̃ strictly ergodic with positive entropy.

To conclude the proof we notice that (S^{1}, f) is a factor of (B(D), σ) , (i.e. there exists a continuous and surjective map π : B(D) → S^{1} such thta π ∘ σ = f ∘ π). It is then easy to prove that, if L is strictly ergodic, then π(L) is strictly ergodic too. moreover π is obtained as the composition of π_{1} : B(D) → B^{+}(D), (the one-sided full shift), and of π_{2} : B^{+}(D) → S^{1}. By definition of the topological entropy (by counting all words of a given length for all lengths), if L is a sub-shift of B(D), π_{1}(L) has the same topological entropy as L. Since π_{2} is almost 2-to-1, by using [4], we conclude that π(L) has the same entropy as L. ■

Definition. 22 Let V be a sub-action and Ω_{0}, Ω_{1} be two disjoint V -irreducible compact invariants sets of Ω(A, f). We say that (Ω_{0}, Ω_{0}) are strongly V - connected if there exists a complete orbit \underline{x} = (x_{n})_{n∈ℤ} included in the V - action set such that α(\underline{x}) ⊂ Ω_{0} and ω(\underline{x}) ⊂ Ω_{1}.

If Ω_{0} and Ω_{1} are periodic orbits of Ω(A, f) and strongly V -connected, such complete orbit \underline{x} satisfying α(\underline{x}) ⊂ Ω_{0} and ω(\underline{x}) ⊂ Ω_{1} could be called an heteroclinic orbit.

Lemma. 23 Let V be a sub-action and let us assume that Ω(A, f) is equal to a finite disjoint union of V -irreducible compact invariant sets {Ω_{i}}_{i=1}^{N}. Let i ≠ j. If (Ω_{i}, Ω_{j}) are V -connected, then there exist a chain (i_{0}, i_{1}, …, i_{r}) of pairewise distinct indexes in {1, …, N} such that i_{0} = i, i_{r} = j and each (Ω_{i_{k−1}}, Ω_{i_{k}l}) are strongly V -connected.

Proof. Step one. We assume that Ω(A, f) can be written as a disjoint union of two ( not necessarily V -irreducible ) compact invariant sets Ω_{0} and Ω_{1}. We assume moreover that there exist p_{0} ∈ Ω_{0} and p_{1} ∈ Ω_{1} which are V - connected. We are going to show the existence of a complete orbit \underline{x} = (x_{n}) included in the V -action set such that x_{0} ∉ Ω(A, f), α(\underline{x}) ⊂ Ω_{0} and ω(\underline{x}) is V -connected to Ω_{1}. We fix once for all two open sets U_{0} and U_{1} containing each Ω_{0} and Ω_{1} such that U_{0} and f(U_{0}) are disjoint from U_{1}. Since (p_{0}, p_{1}) are V -connected, for every ϵ > 0, there exist n_{ϵ} ≥ 1 and p_{ϵ} such that d(p_{0}, p_{ϵ}) < ϵ, f^{n_{ϵ}}(p_{ϵ}) = p_{1} and

V (p_{1}) − V (p_{ϵ}) − ϵ ≤ S_{n}(A − m)(p_{ϵ}) ≤ V (p_{1}) − V (p_{ϵ}).

Let us show, for all 0 ≤ k < l ≤ n,

V ∘ f^{l}(p_{ϵ}) − V ∘ f^{k}(p_{ϵ}) − ϵ ≤ S_{l−k}(A − m) ∘ f^{k}(p_{ϵ}) ≤ V ∘ f^{l}(p_{ϵ}) − V ∘ f^{k}(p_{ϵ}).

On the one hand, because V is a sub-action

S_{l−k}(A − m) ∘ f^{k}(p_{ϵ}) ≤ V ∘ f^{l}(p_{ϵ}) − V ∘ f^{k}(p_{ϵ})

On the other hand

v(p_{1}) − v(p_{ϵ}) − ϵ ≤ S_{n−l}(A − m) ∘ f^{l}(p_{ϵ}) + S_{l−k}(A − m) ∘ f^{k}(p_{ϵ}) + S_{k}(A − m)(p_{ϵ}),

and

S_{k}(A − m)(p_{ϵ}) ≤ V ∘ f^{k}(p_{ϵ}) − V (p_{ϵ}),
S_{n−l}(A − m) ∘ f^{l}(p_{ϵ}) ≤ V (p_{1}) − V ∘ f^{l}(p_{ϵ}).

Combining these inequalities we obtain

V ∘ f^{l}(p_{ϵ}) − V ∘ f^{k}(p_{ϵ}) − ϵ ≤ S_{l−k}(A − m) ∘ f^{k}(p_{ϵ}).

Let s_{ϵ} be the first time the iterates of p_{ϵ} escape U_{0}, that is, f^{k}(p_{ϵ}) ∈ U_{0} for k = 0, 1, …, s_{ϵ} − 1 and f^{s_{ϵ}}(p_{ϵ}) ∉ U_{0}. Let x_{−k}(ϵ) = f^{s_{ϵ}−k}(p_{ϵ}). By a procedure of diagonal extraction, we can find a subsequence of ϵ’s converging to zero such that each x_{−k}(ϵ) converges to some x_{−k}, (remark that s_{ϵ} → ∞ when ϵ → 0). By construction x_{0} ∉ U_{0} ∪ U_{1}, x_{−k} ∈ \overline{U}_{0} for all k = 1, 2, … and the complete orbit \underline{x} = (x_{n}) is included in the V -action set. Moreover any x_{n} is V -connected to p_{1} and α(\underline{x}) ⊂ \overline{U}_{0}. Since α(\underline{x}) is a subset of Ω(A, f), necessarily α(\underline{x}) ⊂ Ω_{0}. Step two. We assume for now on that Ω(A, f) is equal to a disjoint union of irreducible compact invariant sets {Ω_{i}}_{n=1}^{N} and that Ω_{i} is V -connected to Ω_{j}.

Let i_{0} = i. There exists a complete orbit \underline{x}_{i_{0}} not included in Ω(A, f) such that α(\underline{x}_{i_{0}}) ⊂ Ω_{i_{0}} and ω(\underline{x}_{i_{0}}) is V -connected to Ω_{j}. But ω(\underline{x}_{i_{0}}) intersects some Ω_{i_{1}} (after Lemma 14-iv), and i_{1} ≠ i_{0}, otherwise, by irreducibility of Ω_{i_{0}} we would have

α(\underline{x}_{i_{0}}) →^{V} x_{i_{0}}(0) →^{V} ω(\underline{x}_{i_{0}} →^{V} α(\underline{x}_{i_{0}}).

and \underline{x}_{i_{0}}(0) would belong to Ω_{i_{0}}. Either i_{1} = j and we are done or we repeat the previous construction. There exists \underline{x}_{i_{1}} such that α(\underline{x}_{i_{1}}) ⊂ Ω_{i_{1}}, ω(\underline{x}_{i_{1}}) intersects some Ω_{i_{2}} and \underline{x}_{i_{1}}(0) ∉ Ω(A, f). By irreducibility of Ω_{i_{0}} or Ω_{i_{1}}, i_{2} cannot be equal to i_{0} or i_{1}. This process has to stop since the number of irreducible parts is finite. ■

We now show how to construct particuliar subactions.

Definition. 24 Let (x, y) ∈ S^{1} and A : S^{1} → ℝ. We call action potential of A from x to y the following quantity :

S_{A}(x, y) = lim_{ϵ→0} sup{S_{n}(A − m)(x′) | n ≥ 1, d(x′, x) < ϵ, f^{n}(x′) = y}

(Notice that we take the supremum over all paths of length n ≥ 1).

As in the definition of V_{A} in Proposition 11, S_{A}(x, y) is uniformly bounded from above but it may happen that S_{A}(x, y) = −∞. The interesting case takes place when x ∈ Ω(A, f).

Lemma. 25 Let f be a C^{1}-expanding map and A be a C^{α} function. Then the action potential of A satisfies the following properties :

(i) For any sub-action V, for any x, y ∈ S^{1}, S_{A}(x, y) ≤ V (y) − V (x) and x →^{V} y if and only if S_{A}(x, y) = V (y) − V (x).

(ii) For any x, y, z ∈ S^{1}, S_{A}(x, y) + S_{A}(y, z) ≤ S_{A}(x, z), S_{A}(x, x) ≤ 0 and S_{A}(x, x) = 0 if and only if x ∈ Ω(A, f).

(iii) For any x ∈ Ω(A, f), S_{A}(x, .) is a C^{α} sub-action. (In particular, S_{A}(x, y) > −∞ for any y ∈ S^{1}).

Proof. (We only prove part (iii) ). Let x ∈ Ω(A, f) fixed.

Step one. We first show that S_{A}(x, .) never takes the value −∞. Let us define define an approximate action potential :

S_{A}^{ϵ}(x, y) = sup{S_{N}(A − m)(x′) | d(x, x′) < ϵ, f^{n}(x′) = y, n ≥ 1}

where the supremum is taken over all paths of lenght n ≥ 1 sarting within ϵ of x and ending at y. Then S_{A}(x, y) = inf_{ϵ} S_{A}^{ϵ}(x, y). Let ϵ_{0} < \frac{1}{2} and n_{0} ≥ 1, x′ such that d(x, x′) < ϵ_{0}, f^{n_{0}}(x′) = y and S_{n_{0}}(A − m)(x′) is close to S_{A}^{ϵ}(x, y). Since x belongs to Ω(A, f), for every ϵ > 0, tehere exist n ≥ 1, as large as we want λ^{−n}ϵ_{0} < \frac{ϵ}{2}, z such that d(z, x) < \frac{ϵ}{2}, f^{n}(z) = x and |S_{n}(A − m)(z)| < ϵ. We choose z′ in the same inverse branch of lenght n as z such that f^{n}(z′) = x′. Then d(z, z′) ≤ λ^{−n}d(x, x′) < \frac{ϵ}{2}, d(x, z′) < ϵ and

|S_{n}(A − m)(z) − S_{n}(A − m)(z′)| < \frac{Höl_{α}(A)}{λ^{α} − 1}ϵ_{0},

S_{n_{0}+n}(A − m)(z′) ≥ S_{n_{0}}(A − m)(x′) − \frac{Höl_{α}(A)}{λ^{α} − 1}ϵ_{0} − ϵ.

By letting ϵ goes to zero, we obtain

S_{A}(x, y) > S_{A}^{ϵ_{0}}(x, y) − \frac{Höl_{α}(A)}{λ^{α} − 1}ϵ_{0}.

This section shows in addition that, in the definition of S_{A}^{ϵ}(x, y), n can be as large as we want.

Step two. Let (y, z) ∈ S^{1} such that d(y, z) < \frac{1}{2}. Let ϵ > 0 and n ≥ 1 such that λ^{−n} < ϵ. Then there exists x′ such that d(x, x′) < ϵ, f^{n}(x′) = y and S_{A}^{ϵ}(x, y) < S_{n}(A − m)(x′) + ϵ. Let x′′ be the unique n-preimage of z in the same inverse branch as x′. Then d(x′x′′) < ϵ, d(x, x′′) < 2ϵ and

S_{n}(A − m)(x′) ≤ s_{n}(A − m)(x′′) + \frac{Höl_{α}(A)}{λ^{α} − 1}d(y, z).

Combining all these inequalities we obtain

S_{A}(x, y) ≤ S_{A}^{ϵ}(x, y)

≤ S_{n}(A − m)(x′) + ϵ

≤ S_{A}^{2ϵ} + ϵ + \frac{Höl_{α}(A)}{λ^{α} − 1}d(y, z)

≤ S_{A}(x, z) + ϵ + \frac{Höl_{α}(A)}{λ^{α} − 1}[2ϵ + d(y, z)].

and S_{A}(x, .) is C^{α} with α-Hölder constant Höl_{α}(A)/(λ^{α} − 1). Step three. We show that S_{A}(x, .) is a sub-action. For ϵ > 0 we have clearly

S_{A}^{ϵ}(x, y) + (A − m)(y) ≤ S_{A}^{ϵ}(x, f(y)).

By letting ϵ goes to zero we obtain

(A − m)(y) ≤ S_{A}(x, f(y)) − S_{A}(x, y).
■

Proof of Proposition 20. We only prove the implication (ii)⇒(iii). Step one. We first show there exists a C^{β}-sub-action V whose action set contains an invariant compact set K, f(K) = K, which is not equal to a finite union of periodic orbits. Since Ω(A, f) is included en any V -action set, we have the following alternative. Either Ω(A, f) is not equal to a finite union of periodic orbits and we choose K = Ω(A, f) and V is any sub-action or Ω(A, f) is equal to the union of at least two periodic orbits. We choose p, q in two distinct periodic orbits and V (x) = S_{A}(p, x). Then V is a sub-action, by Lemma 25, and p →^{V} q for any q ∈ S^{1}. Lemma 23 tells us there exists a complete orbit \underline{x} included in the V -action set such that α(\underline{x}) and ω(\underline{x}) do not belong to the same periodic orbit; in that case K is the closure of orb(\underline{x}). Step two. As in the proof of theorem 8, we start by writting the cohomological equation for the sub-action founded in step one

A − m = V ∘ f − V − R

where m = m(A, f) and R is a non-negative C^{β}-function which is equal to zero on K. We want to find a sequence of non-negative C^{β}-functions (ϕ_{n}) converging to zero in the uniform topology and uniformly bounded in the C^{β}-topology such that V is still a sub-action of A + ϕ_{n} and the V -action set of A + ϕ_{n} contains a neighborhood of K. Using a remark of T. Bousch [3], we define

ϕ_{n} = min(R, \frac{1}{n}).

Then 0 ≤ ϕ_{n} ≤ R, A+ϕ_{n}−m ≤ V ∘f −V and ϕ_{n} equals R on a neighborhood U_{n} of K (notice that m(A, f) = m(A + ϕ_{n}, f)). By using Lemma 21, U_{n} contains a strictly ergodic compact invariant set L_{n} of positive topological entropy. We finaly choose ψ_{n} non-negative, ψ_{n} = 0 on L_{n} and ψ_{n} > 0 on S^{1} \ L_{n} with small ∥.∥_{β}-norm; A_{n} = A + ϕ_{n} − ψ_{n} is the desired sequence. ■

3 Lyapunov minimizing measures

The idea of the proof of Theorem 2 is first to conjugate f to a fixed map T by preserving the same geometry of the orbits and second to keep the smoothness of f into two conjugacy invariants (A, μ) where A = − ln f′ has pressure zero and μ, the Lebesgue measure, is the unique equilibrium measure associated to (A, f).

We first show in the following lemma that any maps (f, g) in F_{α} are conjugate by bi-hölder maps and that the Hölder exponent can be as close to 1 as we want depending on the C^{1+α}-distance of f and g. Notice that the composition of an α-Hölder map with a β-Hölder map is αβ-Hölder. We use the notation : a ∧ b = min(a, b) and a ∨ b = max(a, b).

Lemma. 26 For simplicity we assume all maps in F_{α} fixe 1 ∈ S^{1}.

(i) For any f, g ∈ F_{α}, there exists a unique orientation preserving homeomorphism θ = θ_{f,g} such that θ(1) = 1 and f ∘θ = θ∘g. The conjugating map is Hölder :

d(θ(x), θ(y)) ≤ C d(x, y)^{γ} ∀ x, y ∈ S^{1}

with Hölder exponent γ = γ(f, g) and γ-Hölder norm C = C(f)∧C(g):

γ(f, g) = inf\{\frac{ln f′(x)}{ln g′(y)} | d(x, y) ≤ \frac{∥\overline{f} − \overline{g}∥_{0}}{λ(f) ∨ λ(g) − 1}\}

C(g) = 2D exp(\frac{Höl_{α}(g′)}{λ(g)^{α} − 1}).

(ii) Let T ∈ F_{α} fixed. Then for any f, g ∈ F_{α}, θ_{f} = θ_{f,T} and θ_{g} = θ_{g,T} satisfy

∥\overline{θ}_{f} − \overline{θ}_{g}∥_{0} ≤ \frac{∥\overline{f} − \overline{g}∥_{0}}{λ(f) ∨ λ(g) − 1}
∥(\overline{θ}_{f})^{−1} − (\overline{θ}_{g})^{−1}∥_{0} ≤ C(T)∥\overline{θ}_{f} − \overline{θ}_{g}∥_{0}^{γ(T,f)∨γ(T,g)}.

where \overline{θ}_{f}, \overline{θ}_{g} : ℝ → ℝ denote the lifts of θ_{f}, θ_{g} with \overline{θ}_{f}(0) = \overline{θ}_{g}(0) = 0.

Proof. Let f ∈ F_{α}, z_{0}^{f} = 1 and z_{1}^{f} … z_{D−1}^{f} the preimages of 1 by f ordered canonically on S^{1} \ {1}. More generally, we call Z(f) the set of preimages of 1 by (f^{n}), n ≥ 0 and code the points in Z(f) by points of the form {z_{i_{1}...i_{n}}^{f}} where i_{n} = 0, 1, …, D−1 and for each n the preimages of 1 by f^{n} are ordered by:

1 = z_{0...000}^{f}, z_{0...001}^{f}, z_{0...010}^{f}, z_{0...011}^{f}, z_{0...100}^{f}, …, z_{1...111}^{f} < 1.

Moreover Z(f) is dense in S^{1}, otherwise any non-empty connected component of S^{1}\Z(f) would be permuted without containing the point 1 and this would contredict the uniform growth of the length.

Since the conjugating map θ_{f,g} has to preserve the order of the preimages, we have necessarily

θ_{f,g}(z_{i}^{g}) = z_{i}^{f}, ∀ i = i_{1} … i_{n}.

Then θ_{f,g} extends uniquely to a homeomorphism which conjugates f and g. Let \overline{f} be the lifting of f fixing 0. We denote by f_{i}^{−1} : [0, 1[→ [0, 1[, i = 0, 1 … D − 1, the inverse branches of \overline{f}:

\overline{f} ∘ f_{i}^{−1}(w) = w + i ∀w ∈ [0, 1[.

Then the points (z_{i_{1}...i_{n}}^{f}) on S^{1} correspond to the points

\overline{z}_{i_{1}...i_{n}}^{f} = f_{i_{1}}^{−1} ∘ ⋯ ∘ f_{i_{n}}^{−1}(0)

on [0, 1[. We first prove that for any w on [0, 1[, n ≥ 1:

∥w_{i_{1}...i_{n}}^{f} − w_{i_{1}...i_{n}}^{g}∥ ≤ \frac{∥\overline{f} − \overline{g}∥_{0}}{λ(f) ∨ λ(g) − 1}

where w_{i_{1}...i_{n}}^{f} = f_{i_{1}}^{−1} ∘ ⋯ ∘ f_{i_{n}}^{−1}(\overline{w}). By induction

|w_{ji_{1}...i_{n}}^{f} − w_{ji_{1}...i_{n}}^{g}|

≤ |f_{j}^{−1}(w_{i_{1}...i_{n}}^{f}) − f_{j}^{−1}(w_{i_{1}...i_{n}}^{g})| + |f_{j}^{−1}(w_{i_{1}...i_{n}}^{g}) − g_{j}^{−1}(w_{i_{1}...i_{n}}^{g})|

≤ \frac{1}{λ(f)}[|w_{i_{1}...i_{n}}^{f} − w_{i_{1}...i_{n}}^{g}| + ∥\overline{f} − \overline{g}∥_{0}].

(The second term is obtained using the estimate

|\overline{f}^{−1}(u) − \overline{g}^{−1}(u)| = |\overline{f}^{−1} ∘ \overline{g} ∘ \overline{g}^{−1}(u) − \overline{f}^{−1} ∘ \overline{f} ∘ \overline{g}^{−1}(u)|

for any u ∈ [0, D[.) In particular, this proves

∥\overline{θ}_{f,T} − \overline{θ}_{g,T} ∥_{∞} ≤ sup_{i=i_{1}...i_{n}} |\overline{θ}_{f,T} (\overline{z}_{i}^{T}) − \overline{θ}_{g,T} (\overline{z}_{i}^{T})| = sup_{i=i_{1}...i_{n}} |\overline{z}_{i}^{f} − \overline{z}_{i}^{g}|

≤ \frac{∥\overline{f} − \overline{g}∥_{0}}{λ(f) ∨ λ(g) − 1}

Assuming that θ_{f} = θ_{f,T} is β-Hölder, we obtain

∥\overline{θ}_{f}^{−1} − \overline{θ}_{g}^{−1}∥_{0} = ∥\overline{θ}_{f}^{−1} ∘ \overline{θ}_{g} ∘ \overline{θ}_{g}^{−1} − \overline{θ}_{f}−1 ∘ \overline{θ}_{f} ∘ \overline{θ}_{g}^{−1}∥_{0}
≤ Höl_{γ}(\overline{θ}_{f}^{−1})∥\overline{θ}_{g} − \overline{θ}_{f}∥_{0}^{γ}.

We now prove that θ = θ_{f,g} is Hölder. We begin by proving that, for all n ≥ 1, for any consecutive preimages of order n:

\overline{z}_{i}^{g} = g_{i_{1}}^{−1} ∘ ⋯ ∘ g_{i_{n}}^{−1}(0), \overline{z}_{i+1}^{g} = g_{i_{1}}^{−1} ∘ ⋯ ∘ g_{i_{n}}^{−1}(1)

we have \overline{z}_{i}^{f} = \overline{θ}(\overline{z}_{i}^{g}) and |\overline{z}_{i}^{f} − \overline{z}_{i+1}^{f}| ≤ |\overline{z}_{i}^{g} − \overline{z}_{i+1}^{g}|^{β}. Indeed

|\overline{z}_{i}^{f} − \overline{z}_{i+1}^{f}| = ∫_{0}^{1} ∏_{k=1}^{n}[\frac{d}{dw}f_{i_{k}}^{−1}](w_{i_{k+1}...i_{n}}^{f}) dw.

Since |w_{i_{k}...i_{n}}^{f} − w_{i_{k}...i_{n}}^{g}| ≤ ∥\overline{f} − \overline{g}∥_{∞}/[λ(f) ∨ λ(g) − 1], we obtain by definition of γ(f, g):

|\overline{z}_{i}^{f} − \overline{z}_{i+1}^{f}| ≤ ∫_{0}^{1} ∏_{k=1}^{n}[\frac{d}{dw}g_{i_{k}}^{−1}]^{γ}(w_{i_{k+1}...i_{n}}^{g}) dw ≤ |\overline{z}_{i}^{g} − \overline{z}_{i+1}^{g}|^{γ}

(We use Jensen’s inequality to extract the power gamma outside the integral since γ(f, g) < 1 for any f ≠ g.) Let \overline{z}_{i}^{g} and \overline{z}_{i+1}^{g} (i = i_{1} … i_{n}) be two consecutive preimages of order n, then [\overline{z}_{i}^{g}, \overline{z}_{i+1}^{g}[ contains D preimages of order n + 1 that we call \overline{z}_{ij}^{g} where ij = i_{i} … i_{n}j and j = 0, 1 … D − 1. We show that for any j = 0, … D − 1:

|\overline{z}_{i}^{g} − \overline{z}_{i+1}^{g}| ≤ \frac{1}{2}C(g)|\overline{z}_{ij}^{g} − \overline{z}_{ij+1}^{g}|.

where C(g) is given in the above Lemma. It is enough to show that the distortion of g_{i_{1}}^{−1} ∘ ⋯ ∘ g_{i_{n}}^{−1} is uniformly bounded by C(g)/2D, that is :

| \frac{\frac{d}{dw}g_{i_{1}}^{−1} ∘ ⋯ ∘ g_{i_{n}}^{−1}(u)}{\frac{d}{dw}g_{i_{1}}^{−1} ∘ ⋯ ∘ g_{i_{n}}^{−1}(v)}| = ∏_{k=1}^{n} |\frac{g′(v_{i_{k}...i_{n}}^{g})}{g′(u_{i_{k}...i_{n}}^{g})}| ≤ exp ∑_{k=1}^{n} |g′(v_{i_{k}...i_{n}}^{g}) − g′(u_{i_{k}...i_{n}}^{g})|

is bounded by C(g)/2D which is a consequence of

|v_{i_{k}...i_{n}}^{g} − u_{i_{k}...i_{n}}^{g}| ≤ λ(g)^{n−k+1}.

We conclude the proof. Given x, y ∈ S^{1} there exist n ≥ 1 and z ∈ [x, y] such that each [x, z] and [z, y] can be covered by an interval of the form [z_{i}^{g}, z_{i+1}^{g}] of order n and such that each [x, z] and [z, y] contains at least an interval of the form [z_{ij}^{g}, z_{ij+1}^{g}] of order n + 1. Then

d(θ(x), θ(z)) ≤ |\overline{z}_{i}^{f} − \overline{z}_{i+1}^{f}| ≤ |\overline{z}_{i}^{g} − \overline{z}_{i+1}^{g}|^{γ}

≤ (\frac{1}{2}C(g))^{β}|\overline{z}_{ij}^{g} − \overline{z}_{ij+1}^{g}|^{β} ≤ \frac{1}{2}C(g)d(x, y)^{γ}

and an equivalent estimate for d(θ(z), θ(y)). ■

In the next lemma, we show that any C^{1+β}-map of the circle can be approximated by smooth maps, in the C^{1+α}-topology, for any α < β (and not in the C^{1+β}-topology). For the need of this paper we need a more precise statement.

Lemma. 27 For any 0 < α < 1, the space F_{α} ∩ C^{∞} of smooth maps of S^{1} is dense in F_{α+} in the C^{1+α}-topology.

Proof. Let f ∈ F_{β} for some β > α, \overline{f} the lift to ℝ of f, (ρ_{n}) a smooth approximation of unity (i.e. ∫ ρ_{n}(t)dt = 1, the support of ρ_{n} is included in ] − \frac{1}{n}, \frac{1}{n}[) and \overline{f}_{n} = \overline{f} ∗ ρ_{n} − \overline{f} ∗ ρ_{n}(0), the convolution of f with ρ_{n}. Then \overline{f}_{n} is a smooth map, \overline{f}_{n}(x + 1) = \overline{f}_{n}(x) + D, \overline{f}_{n}(0) = 0 and λ(f_{n}) ≥ λ(f). The sequence of maps (\overline{f}_{n}) converge to \overline{f} in the C^{1}-topology. In particular, for large n, f_{n} is a smooth covering of degre D of the circle, preserves orientation and satisfies the same constraints for ist derivative. Moreover the β-Hölder semi-norm of f′_{n} is uniformly bounded:

|f′_{n}(x) − f′_{n}(y)| ≤ ∫ |f′(x − t) − f′(y − t)|ρ(t)dt
≤ Höl_{β}(f′)|x − y|^{β}.

By using lemma 17, (f′_{n}) converges to f′ in the C^{α}-topology for any α < β, and (f_{n}) converges to f in the C^{1+α}topology. ■

We now give a short summary of the theory of Thermodynamic Formalism for any T ∈ F_{α}, which are particular expanding Markov maps. An interested reader may find a comprehensive exposition of this theory in the book [20]. Given a potential A : S^{1} → ℝ, we define the Ruelle-Peron-Frobenius opperator L_{A}, acting on the set of countinuous functions, by the following formula:

(L_{A}.h)(x) = ∑_{T (y)=x} h(y)e^{A(y)} ∀x ∈ S^{1}

where h : S^{1} → ℝ is a continuous function and the summation is taken over the D preimages of x. If A is α-Hölder, L_{A} also acts on α-Hölder functions and by duality, L_{A} acts on the set of probability measures by

∫ h d(L_{A}^{∗}.μ) = ∫ L_{A}.h dμ

for any continuous test functions h. For such Markov expanding and mixing map T and for such α-Hölder potentials A, the spectrum σ(L_{A}) of L_{A}, acting on C^{α}-potentials, admits an isolated simple eigenvalue e^{P_{T} (A)} which dominates the whole spectrum. P_{T} (A) is called the pressure of A and the rest of the spectrum has a strictly smaller modulus:

sup_{z≠exp P_{T} (A)}{|z| | z ∈ σ(L_{A})} < exp P_{T} (A).

Moreover there exist a unique probability measure μ_{A}, called the Gibbs measure, and a unique positive α-Hölder function h_{A} such that

L_{A}^{∗}.μ_{A} = e^{P_{T} (A)}μ_{A} and L_{A}.h_{A} = e^{P_{T} (A)}h_{A}.

We also recall that a map T possesses a Jacobian J : S^{1} →]0, +∞[ with respect to μ if

μ(T (U)) = ∫_{U} J dμ

for all Borel set U where T : U → S^{1} is one-to-one. Then e^{P_{T} (A)−A} is the Jacobian of T with respect to μ_{A}. The measure μ_{A}^{equi} = h_{A}μ_{A} is called the equilibrium measure associated to A, is T -invariant and

\frac{h_{A}}{h_{A} ∘ T} exp(P_{T} (A) − A)

is the Jacobian of T with respect to μ_{A}^{equi}.

In the following lemma we define a kind of ”local” chart (U_{T} , Φ_{T}) about any T ∈ F_{α+}. The difficulty comes from the loss of differentiability in the conjugating map θ which is not Lipschitz but γ-Hölder with γ as close to 1 as we want provided U_{T} is small enough.

Lemma. 28 Let 0 < α < β, T ∈ F_{β} and A = − ln T′.

(i) There exists a C^{1+β}-neighborhood Ũ_{T} of T such that − ln f′ ∘ θ_{f,T} is a C^{α}-function depending continuously on f ∈ Ũ_{T}.

(ii) Conversely, there exist a C^{β}-neighborhood U_{T} of A and a continuous map Φ_{T} : U_{T} → F_{α+} such that for every F ∈ U_{T}, f = Φ_{T} (F) is the solution of the equation,

F − P_{T} (F) = − ln f′ ∘ θ_{f,T} .

Moreover U_{T} can be any neighborhood of the form

U_{T} = {F ∈ C^{β} | ∥F ∥_{β} < C and ∥F − A∥_{0} < ϵ}

where C is any positive constant and ϵ = ϵ(C) is sufficiently small.

Proof. Part (i): Since θ_{f} is \frac{ln λ}{ln D}-Hölder, A = P(f) is \frac{ln λ}{ln D}-Hölder too for any f in F_{1}(λ, Λ). If B = P(g), then

|A(x) − B(x)| ≤ |f′ ∘ θ_{f}(x) − g′ ∘ θ_{g}(x)|

(we have used that |f′| > λ > 1), and

|A(x) − B(x)| ≤ ∥f′ − g′∥_{∞} + Λd(θ_{f}(x), θ_{g}(x))

By using lemma 26, we obtain

∥A − B∥_{∞} ≤ ∥f′ − g′∥_{∞} + \frac{Λ}{λ − 1}∥\overline{f} − \overline{g}∥_{∞}.

Moreover the \frac{ln λ}{ln D}-Hölder norm of P(f) is uniformly bounded:

|A(x) − A(y)| ≤ |f′ ∘ θ_{f}(x) − f′ ∘ θ_{f}(y)| ≤ ΛC(λ)d(x, y)^{ln λ/ ln D}.

Continuity of P now follows from lemma 17.

Part (ii): For any C^{β}-function F, exp(P_{T} (F)−F) is the jacobian of T with respect to the Gibbs measure μ_{F}. If θ_{F} : S^{1} → S^{1} is an homeomorphism, exp(P_{T} (F) − F) ∘ θ_{F}^{−1} is the jacobian of f = θ_{F} ∘ T ∘ θ_{F}^{−1} with respect to the push measure (θ_{F})_{∗}μ_{F}. We are therefore looking for a θ_{F} which satisfies (θ_{F})_{∗}μ_{F} = Leb and θ_{F} (1) = 1. The only possibility for θ_{F} is given by the formula

Leb(\overline{θ}_{F} [0, x]) = \overline{θ}_{F} (x) = \overline{μ}_{F} ([0, x]) ∀x ∈ [0, 1[

where \overline{θ}_{F} is the lift of θ_{F} and \overline{μ}_{F} is the corresponding measure on [0, 1[. Since μ_{F} gives positive measure to non-empty open sets and has no mass θ_{F} : S^{1} → S^{1} is an homeomorphism. Let f = θ_{F} ∘T ∘θ_{F}^{−1}, then f is a covering of degree D and preserves the orientation. Moreover J = exp(P_{T} (F)−F ∘θ_{F}^{−1}) is the Jacobian of f with respect to the Lebesgue measure and

\overline{f}(y) − \overline{f}(x) = ∫_{x}^{y} \overline{J}(t) dt

for any 0 < x < y < 1. This shows in particular that f is C^{1} and − ln f′∘θ_{F} = F − P_{T} (F). By lemma 26, θ_{f} = θ_{F}, θ_{F}^{−1} is γ(T, f)-Hölder and its γ-Hölder constant is bounded by C(T). Then f′ is βγ-Hölder and

Höl_{βγ}(f′) ≤ ∥ exp(P_{T} (A) − A∥_{0}Höl_{β}(A)C(T)^{β}.

Let γ^{∗} < 1, close to 1, such that βγ^{∗} > α. Let λ^{∗} < λ(T) < Λ^{∗}, C^{∗} > Höl_{β}(A) and

U_{T} = {F ∈ C^{β} | Höl_{β}(F) < C^{∗} and ∥F − A∥_{0} < ϵ}.

We want to show that, for ϵ small enough,

max_{F ∈U_{T}} exp(P_{T} (F) − F) ≤ Λ^{∗},  min_{F ∈U_{T}} λ(f) = min_{F ∈U_{T}} exp(P_{T} (F) − F) ≥ λ^{∗}

γ(T, f) ≥ γ^{∗}, the βγ^{∗}-Hölder constant of f′ is uniformly bounded and f′ depends continuously with respect to F. Using lemma 17, this would prove the continuity of Φ_{T} (F) with respect to F ∈ U_{T} as a C^{1+α} maps.

We first notice the following estimates:

∥\overline{f} − \overline{g}∥_{0} ≤ ∥f′ − g′∥_{0}

where f′ = exp(P_{T} (F) − F) ∘ θ_{F}^{−1}, g′ = exp(P_{T} (G) − G) ∘ θ_{G}^{−1}. Then we have

∥f′ − g′∥_{0} ≤ Λ^{∗} [2∥F − G∥_{0} + C^{∗}∥(\overline{θ}_{F})^{−1} − (\overline{θ}_{G})^{−1}∥_{0}^{β}]
∥(\overline{θ}_{F})^{−1} − (\overline{θ}_{G})^{−1}∥_{0} ≤ C(T)∥\overline{θ}_{F} − \overline{θ}_{G}∥_{0}^{ln λ^{∗}/ ln Λ^{∗}}

To conclude, it is now enough to prove that \overline{θ}_{F} depends continuously with respect to F ∈ U_{T} in the uniform topology.

Let (F_{n}) be a sequence of β-Hölder functions converging to F in the C^{β}- topology and (f_{n}) the corresponding maps. We show that (\overline{θ}_{F_{n}}) converges to \overline{θ}_{F} uniformly. Restricted to [0, 1], \overline{θ}_{F_{n}} corresponds to the distribution of the measure \overline{μ}_{F_{n}}. Since \overline{θ}_{F}, \overline{θ}_{F_{n}} are increasing and continuous, by using Helly’s theorem, it is enough to show that (\overline{θ}_{F_{n}}) converges pointwise to \overline{θ}_{F} or that (\overline{μ}_{F_{n}}) converges weakly to \overline{μ}_{F} (we use here the fact that \overline{μ}_{A} has no atom).

Let ν be a weak limit of some sub-sequence (μ_{F_{n′}}). Since L_{F_{n}} converges to L_{F} in the uniform topology, for any test funnction h : S^{1} → ℝ:

lim_{n′→∞} ∫ L_{F_{n′}}.h dμ_{F_{n′}} = ∫ L_{F} .h dν

∫ L_{F_{n′}}.h dμ_{F_{n′}} = e^{P (F_{n′})} ∫ h dμ_{F_{n′}}

lim_{n′→∞} e^{P (B_{n′})} ∫ h dμ_{B_{n′}} = e^{P (F)} ∫ h dν.

We thus obtain that ν is solution of L_{F} .ν = e^{P (F)}ν, and by uniqness of Gibbs measures, ν = μ_{F}. Since γ(T, f) depends continuously with respect to f in the C^{1}-topology, γ(T, f) can be made bigger than γ^{∗} for every F ∈ U_{T} as soon as ϵ is choosen small enough. ■

Remark. 29 In the previous lemma, (U_{T} , Φ_{T}) is not a one-to-one parametrization. Although we do not need it, we could have been more precise. The set P = {A ∈ C^{α} | A < 0 and P_{T} (A) = 0} is actually a C^{1}-manifold: for each A ∈ P, P is homeomorphic to an open set in a Banach space T_{A} = {F ∈ C^{α} | F < 0 and ∫ (F − A)dμ_{A}^{equi} = 0}. Indeed if F ∈ P,

\frac{∫ A dμ_{A}^{equi}}{∫ F dμ_{A}^{equi}} F ∈ T_{A} and \frac{∫ A dμ_{A}^{equi}}{∫ F dμ_{A}^{equi}} ≥ 1

where μ_{A}^{equi} is the equilibrium measure associated to A. Conversely, if F ∈ T_{A}, the pressure P is C^{1}-differentiable on the space of C^{α}-potentials (see [17]) and

P′_{T} (A).H = ∫ H dμ_{A}^{equi}

The map p(t, F) = P_{T} (tF) is therefore C^{1} with respect to t and F,

\frac{∂p}{∂t} = ∫ F dμ_{F}^{equi} < 0, p(0, F) = ln D and p(1, F) ≤ 0.

The function p(t, F) is decreasing with respect to t, there exists a unique 0 < δ(F) ≤ 1 such that p(δ(F), F) = 0. By the implicit function theorem, the function F ↦ δ(F) is C^{1}.

Proof of Theorem 2. Let G be the set of maps f in F_{α} admiting a unique Lyapunov minimizing measure supported on a periodic orbit and satisfying the property of countinuously varying support. We first show that G is open in F_{α} and that G_{+} = G ∩ F_{α+} is dense in F_{α+}(λ, Λ).

To show that G is open, we proceed as in the proof of theorem 8 Part (ii). Let f_{0} ∈ G, μ_{0} = \frac{1}{N} ∑_{k=0}^{N−1} δ_{f_{0}^{k}(p_{0})} the unique minimizing periodic measure, N a neighborhood of the orbit orb(p_{0}) = {p_{0}, f_{0}(p_{0}), …, f_{0}^{N}(p_{0}) = p_{0}} and U a neighborhood of f_{0} in the C^{1+α}-topology such that, for any f ∈ U, f has a unique periodic orbit p of period N in N which is also the only f-invariant compact set in N. We note = u_{f} = \frac{1}{N} ∑_{k=0}^{N−1} δ_{f^{k}(p)} the corresponding periodic measure. Thanks to the property of continuously varying support, U can be cosen small enough so that, for any f ∈ U, any Lyapunov minimizing measure μ for f has a support included in N and is therefore equal to μ_{f}. To prove that any f ∈ U also satisfies the property of continuously varying support, we choose (f_{n}) converging to f in the uniform topology and notice that any accumulation set K of the sequence (supp(μ_{f_{n}}) is equal to the unique periodic orbit of f in N.

We now show that G_{+} is dense in F_{α+}. Let T ∈ F_{α+}, then T ∈ F_{β} for some β > α and A = − ln T′ is C^{β}. According to Theorm 8 (and more precisely its proof) there exists a sequence (F_{n}) of β-Hölder functions having a β-Hölder norm uniformly bounded converging to A in the C^{0}-topology and such that F_{n} ∈ G_{+} for all n. Thanks to lemma 28, (Φ_{T} (F_{n})) converges to T as C^{1+α} maps and the set of Lyapunov minimizing measures of Φ_{T} (F_{n}) coincides with the set of maximizing measures of F_{n}.

The proof of the second part of Theorem 2 follows from the equivalence between (i) and (ii) in Proposition 20. ■

NOT YET FINISHED

The next proposition appeared in a slight different form in (see Prop 4 in [16]).

Proposition. 30 Suppose μ_{A}, maximizing A is unique, then μ̂_{t}, when t goes to −∞, converges to μ_{A}. Proof : Suppose μ̂_{t} does not converges to μ_{A}, then for some ϵ > 0 and a subsequence t_{n},

∫ Adμ̂_{t_{n}} < ∫ Adμ_{A} − ϵ.

Take t_{n} negative enough such that t_{n}ϵ + log 2 < 0.

Then

h(μ_{A}) − t_{n} ∫ Adμ_{A} ≥ h(μ_{A}) − t_{n}(∫ Adμ̂_{t_{n}} + ϵ) ≥

h(μ_{A}) − t_{n}ϵ − t_{n} ∫ Adμ̂_{t_{n}} ≥ h(μ̂_{t_{n}}) − log 2 − t_{n}ϵ − t_{n} ∫ Adμ̂_{t_{n}} ≥

h(μ̂_{t_{n}}) − t_{n} ∫ Adμ̂_{t_{n}}.

This is a contradiction with the fact that μ̂_{t_{n}} is equilibrium state for A.

The result above shows that when t goes to infinity, in the problem of the pressure described above, we are putting less and less stressing in the entropy of the measure and more and more emphasys on the integral of the measure.

References

[1] G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc. (2), N 13 (1976) 486-488.

[2] Th. Bousch. Le poisson n’a pas d’arêtes, Preprint.

[3] Th. Bousch. Private communication to the last author.

[4] R. Bowen. Entropy for maps of the interval. J. Topology, 16 (1977), 465–467.

[5] M. J. Dias-Carneiro, On minimizing measures of the action of autonomous Lagrangians, Nonlineariry 8 (1995), 1077-1085

[6] G. Contreras, J. Delgado and R. Iturriaga , Lagrangian flows: the dynamics of globally minimizing orbits - II, Bol. Soc. Bras. Mat. Vol. 28, N.2, 155-196, (1997). Also available via internet at“http://www.ma.utexas.edu/mp arc”.

[7] G. Contreras, R. Iturriaga, G.P. Paternain, M. Paternain. Lagrangian graphs, minimizing measures and Mañé’s critical values. To appear in GAFA. Also available via internet at “http://www.ma.utexas.edu/mp arc”.

[8] A. Fathi. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris, t. 324 Série I, 1043-1046, (1997).

[9] G. Forni, J. Mather, Action minimizing orbits in Hamiltonian systems, Lect. Notes in Math. 1589, Springer Verlag (1994) 92-186.

[10] C. Grillenberger, Constructions of strictly ergodic systems I. given entropy, Z. Wahrscheinlichkeitstheorie verw Geb, 25, (1973) 323-334.

[11] C. Grillenberger, Constructions of strictly ergodic systems II. Ksystems, Z. Wahrscheinlichkeitstheorie verw Geb, 25, (1973) 3335-342.

[12] B.R. Hunt. Maximal local Lyapunov dimension bounds the box dimension of chaotic attractors. Nonlinearity Vol.9 (1996), 845–852.

[13] B.R. Hunt. Optimal orbits of hyperbolic systems. Preprint.

[14] O.M. Jenkinson. Frequency-locking on the boundary of the barycenter set. Preprint.

[15] O.M. Jenkinson. On the barycentre of invariant measures for circle maps. Preprint.

[16] A. Lopes, Dimension spectra and a mathematical model for phase transitions, Advances in Applied Math, Vol 11 , N 4, (1990), 475-502.

[17] R. Mañe, The Hausdorff dimension of horseshoes of difeomorphisms of surfaces, Bol. da Soc. Bras. Mat., Vol 20, N 2, (1990) 1-24.

[18] R. Mañe, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity 9, (1996), 273-310

[19] R. Mañé. Lagrangian Flows: The Dynamics of Globally Minimizing Orbits. In Int. Congress on Dyn. Sys. in Montevideo (a tribute to Ricardo Mañé), F. Ledrappier, J.L. Lewowicz, S. Newhouse eds., Pitman Research Notes in Math. 362 (1996) 120-131. Reprinted in Bol. Soc. Bras. Mat. Vol 28, N. 2, 141-153.

[20] W. Parry and M. Pollicott, Zeta function and the periodic orbit structure of hyperbolic dynamics, Asterisque, Vol 187-188, (1990)

[21] R. Rockafellar, Convex Analysis, Princeton Univ Press, (1970)

[22] Ph. Thieullen, Entropy and the Hausdorff dimension for infinitedimensional dynamical systems, Journal of Dynamics and Differential Equations, vol 4, N 1, (1992), 127-159