{'fraction_non_alphanumeric': 0.06857219900229203, 'fraction_numerical': 0.013105028987461237, 'mean_word_length': 3.1266273506175586, 'pattern_counts': {'":': 0, '<': 43, '<?xml version=': 0, '>': 35, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Lyapunov minimizing measures for expanding maps of the circle

We consider the set of maps f\in\mathcal{F}_{\alpha+} = \cup_{\beta>\alpha} \mathcal{C}^{1+\beta} of the circle which are covering maps of degree D, expanding, \min_{x\in 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 \mu minimizing \int \ln f'\,d\mu over all f-invariant probability measures. We show there exists a set \mathcal{G}_+\subset\mathcal{F}_{\alpha+}, open and dense in the \mathcal{C}^{1+\alpha}-topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the \mathcal{C}^{1+\alpha}-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 analogous quation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8]. We will also present some results on the set of f-invariant measures \mu maximizing \int A\,d\mu for a fixed \mathcal{C}^1-expanding map f and a general potential A, not necessarily equal to -\ln f'.


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: A measure in M(f ) is called a Lyapunov minimizing measure.
The purpose of this paper is to show the following theorem. We recall first that a compact invariant set is said to be strictly ergodic if it is minimal and uniquely ergodic or, equivalently, if it is uniquely ergodic and the support of the unique invariant measure is equal to the compact set itself. (ii) If f ∈ F α+ has a Lyapunov minimizing measure not supported on a finite set of periodic points, then f 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.
In the case when the unique maximizing measure µ is supported by a single orbit, because the map f is expanding, the property of continuously varying support is equivalent to saying there exists a neighborhood U of A in the C α -topology such that, for all B in  As before, the main tool to prove Theorem 8 is the following sub-cohomological equation. For a generalization to smooth Anosov diffeomorphisms, see [23]. 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 subsolution of the Hamilton-Jacobi equation: 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 §2 we prove Theorems 6, 8 and 9 for a fixed expanding maps f ∈ F 0 . We then transfer in §3 the above properties to prove Theorems 2 and 4 for varying functions f . In that case, the potential A is linked to the function by the formula A = −log f and the thermodynamic formalism is used. Analogous results hold for topologically mixing one-sided sub-shifts of finite type in D-symbols and by means of a Markov partition to general mixing expanding maps on a compact set (see [4] for the construction of such a Markov partition).

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. Nonetheless, 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 We claim that R is open in C 0 (S 1 , R) and R ∩ H is dense in H for the · H -topology. The desired residual set will be R = ∩ R ∩ H. We show by contradiction that R is open. If not, one can find A in R , 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 zero and d(µ n , ν n ) ≥ for all n. We may assume by taking a subsequence that (µ n ) and (ν n ) converge toμ andν. Let us prove thatμ ∈ M(A): indeed for every µ ∈ K, and A dµ ≤ A dμ by taking the limit on n. For the same reasonν belongs to M(A). We have obtained a contradiction since d(μ,ν) ≥ .
We now show that R ∩ 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 −1 n (p n ) ≤ 2 −n and we choose n large enough so that 2 −n < . By definition of p n = (p 1 , . . . , p n ) there exists (a 1 , . . . , a n ) ∈ R n such that n i=1 a i p i > n i=1 a i q i ∀q = (q 1 , . . . , q n ) ∈ π n (K 0 ), q = p.
then M 0 (A 1 ) = π −1 n (p n ) has diameter less than . We show that, for small enough 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 zero. We may assume that (µ n ) converges toμ ∈ K \ U. We first show thatμ ∈ 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 dμ. We then show thatμ belongs to M 0 (A 1 ): for every µ ∈ K 0 , Since A 0 dµ n ≤ A 0 dµ, we have obtained A 1 dµ ≤ A 1 dµ n and at the limit We now prove the cocycle Theorem 9. We show that any A ∈ C α is sub-cohomologous 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 f n (I ) = S 1 . Theorem 9 is strongly related to Theorem B of Mañé [17]. Our result is in some sense stronger since we obtain a sub-coboundary defined everywhere (this fact is crucial in the following) whereas Mañé's proof gives a coboundary defined only on the support of the maximizing measure.
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: Proof. We may assume m(A, f ) = 0. We recall that S n A denotes the Birkhoff sum n−1 k=0 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)) < 1 2 and 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 ) < 1 2 then and, by taking the supremum over y and n, we finally obtain Remark 12. We collect here alternative proofs of the existence of a sub-coboundary.
(i) If we choose in Proposition 11 the following definition ofṼ A : we again obtain a sub-coboundary which satisfies in addition the functional equation considered by Bousch [2] and Fathi [8]: In particular (D = 2), for any opposite points x, x (i.e. f (x) = f (x )), one of them belongs to the set has therefore always a non-empty interior and x ∈ ∂ ⇒ x ∈ . A second way to prove the existence of a sub-coboundary is to use Proposition 23(iii) where we define an action potential S A (x, y). A third way is to use the thermodynamic formalism: see Proposition 29(iii). (ii) If A is not equal to a coboundary modulo a constant then the set of α-Hölder (iii) Proposition 11 shows that, in the case when A admits a unique maximizing measure µ, f is uniquely ergodic on supp(µ). This follows from the fact that any measure ν with support contained in the support of µ satisfies A dν = m(A, f ). A maximizing measure for A may not be unique: for instance, when A is of the form In order to prove the property of continuously varying support, we introduce a vocabulary close to what is used in the setting of Lagrangian flows.
Definition 13. Let A be a continuous function and m = m(A, f ).
We then call the set the W -action set (we could have called it the Mañé 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 > 0, there exist z ∈ S 1 and n ≥ 1 such that d(z, x) < , f n (z) = y and there exist z ∈ S 1 and n ≥ 1 such that We denote by (A, f ) the set of non-wandering points with respect to (A, f ).
We will see shortly that (A, f ) is not empty. (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 x = (x n ) n∈Z in S 1 (f (x n ) = x n+1 for all n ∈ Z), ω(x) denotes the compact invariant set of accumulation points of (x n ) n≥0 and α(x) the compact invariant set of accumulation points of (x −n ) n≥0 . We also recall that an -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 ) < 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.  n∈Z is a complete orbit in the W -action set, then α(x) and ω(x) are compact invariant W -irreducible sets and therefore belong to (A, f ). For any

1387
The main ingredients for proving Theorem 8 are given by the following two propositions. Proposition 15(ii) shows in particular that M(A, f ) is equal to the set of all invariant measures whose support stays in (A, f ). PROPOSITION 15. Let A be a Hölder function. Then: In particular, Let x be a point of (A, f ) and > 0. We choose, using the continuity of f and A, i≥0 is a sequence of points in S 1 converging to x , (n i ) i≥0 is 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 zero, 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 zero. We have proved To prove that (A, f ) is included in the W -action set, we introduce the non-negative function converges to zero and therefore R(x) = 0. Part (ii). The fact that the support of any maximizing measure µ is included in (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) in supp(µ), 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 have just shown that supp(µ) is W -irreducible.
Part (iii). We now assume that (A, f ) is W -irreducible for some Hölder sub-action W . Let K be a compact invariant set included in the W -action set. We pick z = (z n ) n∈Z , a complete orbit in K passing through z (i.e. z 0 = z), α ∈ α(z) and ω ∈ ω(z). Then Part (iv). The proof is very similar to part (iii). Let µ be the unique maximizing measure, x, y, points of (A, f ), and x = (x n ) n∈Z , y = (y n ) n∈Z complete orbits passing through x, y, respectively. We define two sequences of probability measures: By taking subsequences, (ω n ) and (α n ) converge to the same measure µ and in particular Proposition 25 shows that A,W may contain invariant compact sets bigger than (A, f ): it may contain heteroclinic orbits connecting two irreducible sets. 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 limit of (µ n ) n≥0 is again a maximizing measure for (A, f ). Indeed, we have for all A, B and for any sub-sequence (µ n ) n ≥0 converging to some measure µ, Step two. Let K n = (A n , f ) and K be a limit set in the Hausdorff topology of some sub-sequence (K n ) n ≥0 . Thanks to Propositions 11 and 15(i), 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: 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 subsequence (V n ) n ≥0 which converges uniformly to an α-Hölder function V . We have thus 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 (A, f ) is V -irreducible by minimality, we know from Proposition 15(iii) that K has to be included in (A, f ) and therefore has to be equal to (A, f ).
Step three. We assume now that (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 ). Thanks to Proposition 15(ii), K n is included in (A n , f ), and any accumulation set of (K n ) n≥0 is contained in (A, f ) and equals (A, f ) by minimality. Any weak-accumulation measure has a support included in (A, f ) and is therefore equal to µ by unique ergodicity. 2 We are now able to prove the first part of Theorem 8.

Proof of Theorem 8(i).
We actually prove a little more. Let Z be the set of α-Hölder is uniquely ergodic is also generic according to Proposition 10. Therefore, Z ∩ R is generic and any A ∈ Z ∩ R satisfies the property of continuously varying support as is shown in Proposition 16(ii). For every > 0, we define Let us first notice that Z = ∩ Z . In order to show that Z is dense, for any A ∈ C α , we choose some K ⊂ (A, f ) minimal. Then we can find φ ∈ C α such that φ = 0 on K, φ > 0 everywhere outside K and such that We now show that Z contains Z in its interior. Let A ∈ Z, then there exists n ≥ 0 such that for every y ∈ U . By Proposition 16(i), any B close to A (in the C α -topology) satisfies (B, f ) ⊂ U and therefore is included in Z . 2

G. Contreras et al
The strategy of the proof of Theorem 8(ii) is to find a periodic orbit well distributed and closed to (A, f ). We begin by proving a lemma of approximation in the C α -topology for functions having a better regularity C β , β > α.
LEMMA 17. Let 0 < α < β < 1. For any β-Hölder functions A, In particular, if (A n ) is a sequence of β-hölder functions uniformly bounded in the C βtopology, if (A n ) converges to zero in the C 0 -topology, then (A n ) converges to zero in the C α -topology.
Proof. The two estimates 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 . 2 Although the following is a very standard lemma, we need a precise estimate of the shadowing constant in the proof of Lemma 19 and so we give a proof.
LEMMA 18. Let f be C 1 -expanding, λ = λ(f ) and N be such that λ N > 2. Then for any x ∈ S 1 such that d(x, f N (x)) is less than 1 4 , there exists a periodic point p of period N such that 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 pre-image of is therefore a periodic point of period N and the length of the interval ]p, x 1 [ is bounded by In particular, this length is bounded by 1 2 The two inequalities combined end the proof.
Proof. For each K > 0 we define Since no periodic orbit belongs to , d(K) > 0 and tends to zero when K tends to infinity. We now choose K so that λ K > 5 and Let be small enough such that < 1 3 d(K). We first exhibit an almost closed orbit for all 0 ≤ i ≤ N. We now show that p is the good candidate. For and we obtain in this case d(f i (p), f j (p)) > > η . For and we obtain

G. Contreras et al
Proof of Theorem 8(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 = n −1 n−1 k=0 δ f k (p) be 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 C 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 ⊂ 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 (A, f ) is reduced to a single periodic orbit. From Proposition 16(ii), we know that A will satisfy the property of continuous varying support. We actually prove the existence of a sequence (φ ) of C β -functions having a β-norm, uniformly bounded and converging to zero in the uniform topology when goes to zero.
Since R 1 = 0 on (A 1 , f ) and R 1 (q) ≤ C 1 d(q, (A 1 , f )) β , we obtain for all q ∈ orb(p) We now define for each q on the orbit a localized function φ ,q . The functions {φ ,q } q∈orb(p) have disjoint support; they satisfy 0 ≤ φ ,q ≤ R 1 everywhere and φ 1,q (q) = R 1 (q) for all q ∈ orb(p), by the following formula where D 1 = (2/η) β C 1 . Clearly D 1 ≥ C 1 and by the Hölder property of R 1 we have φ ,q ≤ R 1 . The support of each φ ,q is included in the ball of center q and radius On the one hand, the Höl β -semi-norm of φ ,q is uniformly bounded, independently of , On the other hand, the · 0 -norm of φ ,q tends to zero when tends to zero, We have thus obtained a non-negative function for all points q ∈ orb(p). By using Lemma 17, φ ∈ C β has small · α -norm. Moreover, m(A 1 + φ , f ) = m 1 and (A 1 + φ , f ) contains both (A 1 , f ) and orb(p). Indeed > 0 and z, n > 0 have been chosen such that d(x, z) < , d(x, f n (z)) < and n−1 k=0 R • f k (z) < , then n−1 k=0 φ • f k (z) < and we have shown that x ∈ (A 1 + φ , f ). Actually, although we do not need it, we can show that We now choose ψ ≥ 0, ψ ∈ C β , with small · β -norm such that ψ = 0 on orb(p) and ψ > 0 elsewhere. Then A = A 1 + φ − ψ ∈ C β is close to A 0 in the C α -topology and admits a unique maximizing measure supported on orb(p): Proof of Theorem 8(iii). We actually give a criterion to decide whether (A, f ) can stay stably a finite union of periodic points. The proof of Theorem 8 is a direct consequence of the following proposition.
PROPOSITION 20. Let f be a C 1 -expanding map, 0 < α < β, and A be a β-Hölder function. Then one of the following cases occurs.
(i) 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 ergodic and has positive topological entropy.

(ii) (A, f ) is a finite union of periodic points and there exists a C β neighborhood U of A such that, for all B ∈ U, (B, f ) ⊂ (A, f ).
Before proving Proposition 20, we need the following lemma.

LEMMA 21. Let K be a compact set (not necessarily invariant). Then K admits a basis of closed neighborhoods U such that
possesses the following property: (i) either U has positive entropy and U contains a strictly ergodic invariant compact set of positive entropy; or (ii) U has zero entropy and the set of periodic points belonging to U is finite.
Proof. We first assume that K is a compact set of the one-sided full shift B + (D). Let V be an open set containing K and C 1 , . . . , C N , a finite union of cylinders covering B + (D) \ V and disjoint from K. We may assume the cylinders have all the same length. Let U = B + (D) \ (C 1 ∪ · · · ∪ C N ). Then U is a closed neighborhood of K and is topologically conjugate to a one-sided Markov chain. If the entropy of U is equal to zero, the set of periodic points in U is finite. If the entropy of U is positive, we use Grillenberger's theorems [10] and [11] to construct a strictly ergodic compact set of positive entropy inside U . In the general case, let K be a compact set of S 1 and V a neighborhood of K . We denote by π : B + (D) → S 1 the canonical extension of S 1 . Let K = π −1 (K ), V = π −1 (V ), C 1 . . . C N , U as before and U = π(U). Then U is a closed neighborhood of K (π(C i ) is disjoint from K ) included in V . Moreover, π( U ) ⊂ U and a point x in U which does not belong to π( U ) is necessarily a pre-image of one. Modulo a countable set in U and in U , π is a bijection, U , U have the same topological entropy and the set of periodic points of U (except maybe one) is equal to the projection of the set of periodic points of U .
Finally, if L is a strictly ergodic invariant compact set, π(L) is also strictly ergodic and L and π(L) have the same topological entropy. 2 Proof of Proposition 20.
Step one. Assume there exists a C β sub-action V such that, for any closed neighborhood U of the V -action set the invariant set U has positive topological entropy. As in the proof of Theorem 8(ii), we start by writing the cohomological equation for the sub-action V , 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 subaction of A + φ n and the V -action set of A + φ n contains a neighborhood of K. We define φ n = min R, 1 n . + φ n , f )). By using Lemma 21, U n contains a strictly ergodic compact invariant set L n of positive topological entropy. We finally choose ψ n non-negative, ψ n = 0 on L n and ψ n > 0 on S 1 \ L n with small · β -norm. We have just proved that A n = A + φ n − ψ n converges to A uniformly with bounded C β -norm and that each (A n , f ) is strictly ergodic and has positive entropy.
Step two. Assume that for every C β sub-action V there exists a neighborhood U of K such that U has zero entropy.
We first show that (A, f ) is a finite union of periodic points. Let V be any sub-action. Then, by Lemma 21, there exists a neighborhood U of K such that U contains only a finite number of periodic points. If a periodic point belongs to K, it actually belongs to (A, f ). We can therefore choose U so that all periodic points in U are in (A, f ). Conversely, if x is a point in (A, f ), then x is a limit of periodic points z n of period p n such that S p n (A − m)(z n ) tends to zero. Then S p n R(z n ) also tends to zero and the orbit of z n has to stay closer and closer to K. For n large enough, z n belongs to U and x is a periodic point.
Let (B n ) be a sequence of C β functions, with bounded C β -norm, converging to A in the C α -topology. We want to show that (B n , f ) ⊂ (A, f ) for n sufficiently large. As in step two of the proof of Proposition 16, for each B n we choose a sub-action V n and we may assume (maybe by taking a sub-sequence) that (V n ) converges to some sub-action V of class C β and that the V n -action set K n converges to some compact set included in the V -action set. We choose a neighborhood U as before so that the only periodic points included in U are actually in (A, f ). For n large enough, (B n , f ) ⊂ K n ⊂ U and by the same argument as before (B n , f ) has to be included into the closure of the set of periodic points of U . That is, for large n, (B n The rest of this section is independent of Theorem 8. The reader interested just in the minimization of Lyapunov measures can go directly to §3. We first show how to construct other sub-actions from a function S A (x, y) that we call the action potential. We then define the notion of a V -heteroclinic orbit which is stronger than the notion of V -connection and apply it to the case where there is a finite number of irreducible sets.

Definition 22.
Let (x, y) ∈ S 1 and A : S 1 → R. We call the action potential of A from x to y the following quantity: (notice that we take the supremum over all paths of length n ≥ 1 ending at y and beginning within of x).
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 ∈ (A, f ).
PROPOSITION 23. 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: Proof. (We only prove part (iii).) Let x ∈ (A, f ) be fixed.
Step one. We first show that S A (x, ·) never takes the value −∞. Let us define an approximate action potential where the supremum is taken over all paths of length n ≥ 1 starting within of x and ending at y. Then S A (x, y) = inf S A (x, y). Let 0 < 1 2 and n 0 ≥ 1, x such that d(x, x ) < 0 , f n 0 (x ) = y and S n 0 (A − m)(x ) is close to S A (x, y). Since x belongs to (A, f ), for every > 0, there exist n ≥ 1, large enough so that λ −n 0 < /2, z such that d(z, x) < /2, f n (z) = x and |S n (A − m)(z)| < . We choose z in the same inverse branch of length n as z such that f n (z ) = x . Then d(z, z ) ≤ λ −n d(x, x ) < /2, d(x, z ) < and By letting go to zero, we obtain This section shows in addition that, in the definition of S A (x, y), n can be as large as we want.
Step three. We show that S A (x, ·) is a sub-action. For > 0 we have clearly f (y)).
By letting go to zero we obtain From the first part of Proposition 23, we notice that, if an invariant compact set K is irreducible for two sub-actions V and W , then V − W is constant on K. The last part of the proposition shows that, for any x ∈ (A, f ), there exists a sub-action V (y) = S A (x, y) such that any y ∈ S 1 is V -connected to x (x V → y). We now define a stronger notion of V -connection.
Definition 24. Let V be a sub-action and 0 , 1 be two disjoint V -irreducible compact invariants sets of (A, f ). We say that ( 0 , 1 ) are strongly V -connected if there exists a complete orbit x = (x n ) n∈Z included in the V -action set such that α(x) ⊂ 0 and ω(x) ⊂ 1 .
If 0 and 1 are periodic orbits of (A, f ) and strongly V -connected, such a complete orbit x satisfying α(x) ⊂ 0 and ω(x) ⊂ 1 could be called a heteroclinic orbit.

Then each i is a maximal V -irreducible invariant compact set and cannot be a disjoint union of two invariant compact sets. Let
Proof.
Step one. Let K be an invariant compact set and assume that K can be written as a disjoint union of two (not necessarily V -irreducible) compact invariant sets K 0 and K 1 . We assume, moreover, that there exist p 0 ∈ K 0 and p 1 ∈ K 1 which are V -connected. We show that, if U 0 and U 1 are disjoint open sets containing K 0 and K 1 , there exists x 0 ∈ K 0 ∪ K 1 and a complete orbit x included in the V -action set going through x 0 such that α(x) ⊂ U 0 and p 0 Since (p 0 , p 1 ) are V -connected, for every > 0, there exist n ≥ 1 and p such that d(p 0 , p ) < , f n (p ) = p 1 and Let us show, for all 0 ≤ k < l ≤ n, and On the one hand, because V is a sub-action On the other hand, Combining these inequalities we obtain Let s be the first time the iterates of p escape U 0 , that is, f k (p ) ∈ U 0 for k = 0, 1, . . . , s and f s +1 (p ) . By a procedure of diagonal extraction, we can find a subsequence of 's converging to zero such that each x −k ( ) converges to some x −k (note that s → ∞ when → 0). By construction, Step two. We show that a maximal (for the inclusion) V -irreducible compact invariant set K cannot be equal to a disjoint union of two invariant compact sets K 0 and K 1 . Otherwise, we choose p 0 ∈ K 0 , p 1 ∈ K 1 and by assumption (p 0 , p 1 ) are V -connected in both directions. Thanks to step one there exist complete orbits x and y included in the V -action set such that x and y do not belong to K and The set K 0 ∪ orb(x) ∪ K 1 ∪ orb(y) is compact, invariant and V -irreducible, which contradicts the maximality of K.
Step three. We assume from now on that (A, f ) is equal to a disjoint union of irreducible compact invariant sets { i } N i=1 and that i is V -connected to j . Let i 0 = i. There exists a complete orbit x i 0 not included in (A, f ) such that α(x i 0 ) ⊂ i 0 and ω(x i 0 ) is V -connected to j (we use the fact that α(x i 0 ) can be chosen in a neighborhood of i 0 and has to be included in (A, f )). But ω(x i 0 ) is included in (A, f ) and has to be included into some i 1 (thanks to step two ω(x i 0 ) cannot intersect two i ). Necessarily i 1 = i 0 , otherwise, by irreducibility of i 0 we would have and x i 0 (0) would belong to i 0 . Either i 1 = j and we are done or we repeat the previous construction. There exists x i 1 such that α(x i 1 ) ⊂ i 1 , ω(x i 1 ) is contained in some i 2 and x i 1 (0) ∈ (A, f ). By irreducibility of i 0 or 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. 2

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 transfer the smoothness of f into two 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 Proof. To simplify, we denote µ t = µ equi tA . For any f -invariant measure µ We obtain, therefore, for any maximizing measure µ ∈ M(A, f ) which shows the first assertion. On the one hand, On the other hand, µ → h(µ) is upper semi-continuous and To simplify again, we denote L t = L tA−P T (t A) . For every n, define V n t by L t 1 = exp(tV n t ) for t > 0 and V n 0 = 0 by convention. Since L t leaves invariant the following compact set of functions (for the uniform topology) we obtain for all t ≥ 0 and for all n ≥ 0 Höl α (V n t ) ≤ Höl α (A)/(λ α − 1).
Since (L n t 1) n≥0 converges uniformly to h t = h tA−P T (A) > 0, (V n t ) n≥0 converges uniformly to some V t with the same Hölder coefficient Höl α (A)/(λ α − 1). The eigenfunction h t = exp(tV t ) is a solution of L t h t = h t or L t (h t /h t • T ) = 1. We first obtain (for t > 0) The definition of pressure implies tm(A) ≤ P T (tA) ≤ h top (T ) + tm(A). By letting t go to infinity, any accumulation limit V of (V t ) is a sub-action. Since tR t satisfies more precisely the functional equation, exp(−tR t (x 1 )) + · · · + exp(−tR t (x D )) = 1 for any x 1 , . . . , x D with the same image. When t goes to infinity, a sub-sequence of (R t ) t>0 In the following lemma we define a kind of 'local' chart (U T , 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 one as we want provided U T is small enough.
Moreover, U T can be any neighborhood of the form where C is any positive constant and = (C) is sufficiently small.
Proof. Part (i). Since θ f is γ (f, T )-Hölder and can be as close to one as desired, we choose γ * < 1 such that βγ * > α. We also choose C * > Höl β (T ), λ * < λ(T ), * > (T ) and For any f ∈Ũ T , the βγ * -Hölder norm of F = P(f ) is bounded by For any f , g inŨ T , F = P(f ), G = P(g), (we have used the fact that |f | > λ > 1), and by using Lemma 26 we obtain 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 : F 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 where θ F is the lift of θ F and 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 : F , then f is a covering of degree D and preserves the orientation. Moreover, is the Jacobian of f with respect to the Lebesgue measure and for any 0 < x < y < 1. This shows in particular that f is C 1 and By Lemma 26 (with β instead of α), θ f = θ F , θ −1 F is γ (T , f )-Hölder and its γ -Hölder constant is bounded by C(T ). Then f is βγ -Hölder and Let γ * < 1, close to 1, such that βγ * > α. Let C * > Höl β (A). Let λ * < * such that λ * < λ(T ) and * > (T ) = max x∈S 1 T (x). Define for small , U T = {F ∈ C β | Höl β (F ) < C * and F − A 0 < }. We also get γ (T , f ) ≥ γ * and a uniform upper bound for the βγ * -Hölder constant of f . If we can show that f depends continuously with respect to F in the uniform topology, using Lemma 17, we prove the continuity of T (F ) with respect to F ∈ U T as a C 1+αmaps.
We first notice the following estimate: To conclude, it is now enough to prove that 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 (θ F n ) converges to θ F uniformly. Restricted to [0, 1], θ F n corresponds to the distribution of the measure µ F n . Since θ F , θ F n are increasing and continuous, by using Helly's theorem, it is enough to show that (θ F n ) converges pointwise to θ F or that (µ F n ) converges weakly to µ F (we use here the fact that 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 function 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 uniqueness 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 larger than γ * for every F ∈ U T as soon as is chosen small enough. 2 Remark 31. In the previous lemma, (U T , 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 an embedded 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 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 The function p(t, F ) is decreasing with respect to t, and 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 .

P T (A).H = H dµ
Proof of Theorem 2. Let G be the set of maps f in F α admitting a unique Lyapunov minimizing measure supported on a periodic orbit and satisfying the property of continuously varying support. We first show that G is open in F α and that G + = G ∩ F α+ is dense in F α+ (λ, ).
To show that G is open, we proceed as in the proof of Theorem 8(ii). Let f 0 ∈ G, let be the unique minimizing periodic measure, N a neighborhood of the orbit orb(p 0 ) = {p 0 , f 0 (p 0 ), . . . , f N 0 (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 that be the corresponding periodic measure. Thanks to the property of continuously varying support, U can be chosen 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 Theorem 8 (and more precisely its proof) there