Henk Bruin

Renaud Leplaideur

RENORMALIZATION, THERMODYNAMIC FORMALISM AND QUASI-CRYSTALS IN SUBSHIFTS

We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points V of a renormalization operator R acting on the space of potentials are such that the pressure function γ → P(-γV ) exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of R, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.

1. Introduction 1.1. Background. Phase transitions are a central theme in statistical mechanics and probability theory. In the physics/probability approach the dynamics is not very relevant and just emerges as a by-product of the invariance by translation. The main difficulty is the geometry of the Z d lattice. Considering an interacting particle systems such as the Ising model (see e.g. [START_REF] Gallivotti | Ising model and Bernoulli schemes in one dimension[END_REF][START_REF] Georgii | Gibbs measures and phase transitions[END_REF]), it is possible to find a measure (called Gibbs measure) that maximizes the probability of obtaining a configuration with minimal free energy associated to a Hamiltonian. This is done considering a finite box and fixing the conditions on its boundary. Then letting the size of the box tend to infinity, the sequence of Gibbs measures have a set of accumulation points. If this set varies non-continuously with respect to the parameters (including the temperature), then the system is said to exhibit a phase transition.

In contrast, the time evolution of the system is the central theme in dynamics systems. The theory of thermodynamic formalism has been imported into hyperbolic dynamics in the 70's, essentially by Sinai, Ruelle and Bowen. Gradually, authors started to extend this theory to the non-uniformly hyperbolic case, sometimes applying inducing techniques that are also important in this paper. Initially, phase transitions have been less central in dynamical systems, but the development of the theory of ergodic optimization since the 2000's has naturally led mathematicians to introduce (or rather rediscover) the notion of ground states. The question of phase transitions arises naturally in this context. Note that vocabulary used in statistical mechanics is sometimes quite different from that used in dynamical system. What in statistical mechanics vocabulary is called a "freezing" transition, such as occur in Fisher-Felderhof models (see e.g. [START_REF] Fisher | On discontinuity of the pressure[END_REF]), corresponds in the mathematical vocabulary to the Manneville-Pomeau map or the shift with Hofbauer potential (see e.g. [START_REF] Wang | Statistical physics of temporal intermittency[END_REF] or [26, Exercise 5.8 on page 98] and also [START_REF] Gallavotti | Zeta functions and basic sets[END_REF]).

Renormalization is an over-arching theme in physics and dynamics, including thermodynamic formalism, see [START_REF] Chazottes | Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials[END_REF] for modern results in the direction. The system that we study in this paper is related to cascade of doubling period phenomenon and the infinitely renormalizable maps à la Feigenbaum-Coullet-Tresser, which is on the boundary of chaos (see e.g. [START_REF] Schuster | Deterministic Chaos[END_REF]). Instead of the freezing transitions, the system has its equilibrium state (at phase transition) supported on a Cantor set rather than in a fixed point or a periodic orbit. Stated in physics terminology, we prove that it is possible to reach a quasi-crystal as a ground state before temperature zero by freezing a dynamical system (see Theorems 4 and 5). This issue is related to a question due to van Enter (see [START_REF] Van Enter | Breaking of periodicity at positive temperatures[END_REF]). The original question was for Z 2 -actions, but we hope that ideas here may be exported to this more complicated case.

Returning to the mathematical motivation, the present paper takes the work of [START_REF] Baraviera | The potential point of view for renormalization[END_REF] a step further. We investigate the connections between phase transition in the full 2-shift, renormalization for potentials, renormalization for maps (in complex dynamics) and substitutions in the full 2-shift. Here the attractor of renormalization is a Cantor set, rather than a single point, and its thermodynamic properties turn out to be strikingly different.

We recall that Bowen's work [START_REF] Bowen | Symbolic dynamics for hyperbolic systems, Recent advances in topological dynamics[END_REF] on thermodynamic formalism showed that every subshift of finite type with Hölder continuous potential φ admits a unique equilibrium state (which is a Gibbs measure). Moreover, the pressure function γ → P(-γφ) is real analytic and there are no phase transitions. This is also known as the Griffiths-Ruelle theorem. Hofbauer [START_REF] Hofbauer | Examples for the nonuniqueness of the equilibrium state[END_REF] was the first (in the dynamical systems world) to find continuous non-Hölder potentials for the full two-shift (Σ, σ) allowing a phase transition at some t = t 0 .

A geometric interpretation of Hofbauer's example leads naturally to the Manneville-Pomeau map f MP : [0, 1] → [0, 1] defined as

f MP (x) = x 1-x if x ∈ [0, 1 2 ], 2x -1 if x ∈ ( 1 2 , 1],
with a neutral fixed point at 0. This map admits a local renormalization ψ(x) = x 2 which satisfies (1)

f 2 MP • ψ(x) = ψ • f MP (x) for all x ∈ [0, 1 2
].

If we differentiate Equation [START_REF] Avila | Hausdorff dimension and conformal measures of Feigenbaum Julia sets[END_REF], take logarithms and subtract log ψ ≡ log 1 2 from both sides of the equality, we find [START_REF] Baraviera | The potential point of view for renormalization[END_REF] log

|f MP | = log |f MP | • f MP • ψ(x) + log |f MP | • ψ(x).
Passing to the shift-space again (via the itinerary map for the standard partition {[0, 1 2 ], ( 1 2 , 1]}), we are naturally led to renormalization in the shift. Of prime importance are the solutions of the equation

(3) σ 2 • H = H • σ,
which replaces the renormalization scaling ψ in [START_REF] Avila | Hausdorff dimension and conformal measures of Feigenbaum Julia sets[END_REF]. Equation (2) leads to an operator R defined by

R(V ) = V • σ • H + V • H.
In [START_REF] Baraviera | The potential point of view for renormalization[END_REF], the authors investigated the case of the substitution

H MP : 0 → 00, 1 → 01,
which has a unique fixed point 0 ∞ , corresponding to the neutral fixed point 0 of f MP . In [START_REF] Baraviera | The potential point of view for renormalization[END_REF], the map H MP was not presented as a substitution but we emphasize here (and it is an improvement because it allows more general studies) that it indeed is; more generally, any constant-length k substitution solves Equation (3) (with σ k instead of σ 2 ). It is also shown in [START_REF] Baraviera | The potential point of view for renormalization[END_REF] that the operator R fixes the Hofbauer potential

V (x) := log n + 1 n if x ∈ [0 n 1] \ [0 n+1 1], n > 0.
Moreover, the lift of log f MP belongs to the stable set of the Hofbauer potential. This fact is somewhat mysterious because the substitution H MP is not the lift of the scaling function ψ : x → x/2.

In this paper we focus on the Thue-Morse substitution; see [START_REF] Bowen | Symbolic dynamics for hyperbolic systems, Recent advances in topological dynamics[END_REF] for the definition. It is one of the simplest substitutions satisfying the renormalization equality (3) and contrary to H MP , the attractor for the Thue-Morse substitution, say K, is not a periodic orbit but a Cantor set. Yet similarly to the Manneville-Pomeau fixed point, σ : K → K has zero entropy and is uniquely ergodic. This is one way to define quasi-crystal in ergodic theory.

The thermodynamic formalism for the Thue-Morse substitution is much more complicated, and more interesting, than for the Manneville-Pomeau substitution. This is because Cantor structure of the attractor admits a more intricate recursion behavior of nearby points (although it has zero entropy) characterized by what we call "accidents" in Section 2.3, which are responsible for the lack of phase transitions for the "good" fixed point for R, This allows much more chaotic shadowing than when the attractor of the substitution is a periodic orbit. We want to emphasize here that our results are extendible to more general substitutions, but to get the main ideas across, we focus on the Thue-Morse shift in this paper.

Let K = ∪ n σ n (ρ 0 ) = ∪ n σ n (ρ 1
) be the corresponding subshift of the full shift (Σ, σ) on two symbols. The renormalization equation (3) holds in Σ: H • σ = σ 2 • H, and we define the renormalization operator acting on functions V : Σ → R as

(RV )(x) = V • σ • H(x) + V • H(x).
We consider the usual metric on Σ:

d(x, y) = 1 2 n if n = min{i 1 : x i = y i }.
This distance is sometimes graphically represented as follows:

x 0 = y 0 d d n -1 x n-1 = y n-1 y x Figure 1.
The sequence x and y coincide for digits 0 up to n -1 and then split.

Note that d(H n x, H n y) = d(x, y) 2 n : if x and y coincide for m digits, then H n (x) and H n (y) coincide for 2 n m digits.

The first two results deal with the continuous fixed points for the renormalization operator R. The main issue is to determine fixed points and their weak stable leaf, namely the potentials attracted by the considered fixed point by iterations of R.

The second series of results deals with the thermodynamical formalism; we study if some class of potentials related to weak stable leaf of the fixed points, exhibit a phase transition. In particular, Theorem 5 is related to a question of Van Enter et al. (see e.g. [START_REF] Van Enter | Breaking of periodicity at positive temperatures[END_REF][START_REF] Van Enter | Nonperiodic long-range order for fast-decaying interactions at positive temperatures[END_REF]) asking whether it is possible to reach a quasi-crystal by freezing a system before zero temperature.

The last result (Theorem 6) returns to the geometrical dynamics and shows the main difference between the Thue-More case and the Manneville-Pomeau case. Due to the Cantor structure of the attractor of the substitution, there exist non-continuous but locally constant (on K up to a finite number of points) fixed points for R. As the Hofbauer potential represents the logarithm of the derivative of an affine approximation of the Manneville-Pomeau map, one of these potentials, V u , represents the logarithm of the derivative of an affine approximation to the Feigenbaum-Coullet-Tresser map f f eig : C → C. The main difference with the Manneville-Pomeau case is that here, V u has no phase transition whereaslog |f f eig | has.

1.2.1. Results on continuous fixed points for R. Define the one-parameter family of potentials

(5) U c =    c on [01], -c on [10],
0 on [00] ∪ [START_REF] Epstein | New proofs of the existence of the Feigenbaum functions[END_REF].

It is easy to verify that U c is a fixed point of R. Given a fixed function V : Σ → R, the variation on k-cylinders Var k (V ) is defined as

Var k (V ) := max{|V (x) -V (y)|, x j = y j for j = 0, . . . , k -1}. The condition ∞ k=1 Var k (W ) < ∞ holds if e.g. W is Hölder continuous. Theorem 1. If W is a continuous fixed point of R on K such that ∞ k=1 Var k (W ) < ∞, then W = U c for c = W (ρ 0 ).
As for the Hofbauer case, we produce a non-negative continuous fixed point for R with a well-defined weak stable set1

Theorem 2. There exists a unique function V , such that V = lim m→∞

1 m n-1 k=0 R k V for every continuous V satisfying V (x) = 1 n + o( 1 n ) if d(x, K) = 2 -n . Moreover V is R-invariant, continuous and positive except on K: 1 2n V (x) 1 n-1 if d(x, K) = 2 -n .
1.2.2. Results on Thermodynamic Formalism. We refer to Bowen's book [START_REF] Bowen | Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms[END_REF] for the background on thermodynamic formalism, equilibrium states and Gibbs measures in Σ. However, in contrast to Bowen's book, our potentials are not Hölder-continuous.

For a given potential W : Σ → R, the pressure of W is defined by

P(W ) := sup{h µ (σ) + W dµ},
where h µ (σ) is the Kolmogorov entropy of the invariant probability measure µ. The supremum is a maximum in Σ whenever W is continuous. Any measure realizing this maximum is called an equilibrium state. We want to study the regularity of the function γ → P(-γW ). For simplicity, this function will also be denoted by P(γ). If P(γ) fails to be analytic, we speak of a phase transition. We are in particular interested in the special phase transition as γ → ∞: easy and classical computations show that P(γ) has an asymptote of the form -aγ + b as γ → ∞. By an ultimate phase transition we mean that P(γ) reaches its asymptote at some γ . In this case, there cannot be another phase transition for larger γ, hence ultimate. Then, by a convexity argument, P(γ) = -aγ + b for any γ γ . One of the main motivations for studying ultimate phase transitions is that the quantity a satisfies a = inf W dµ, µ is a shift-invariant probability measure .

An example of an ultimate phase transition for rational maps can be found in [START_REF] Makarov | On thermodynamics of rational maps. II. Non-recurrent maps[END_REF]. The Manneville-Pomeau map is another classical example.

Theorem 3 (No phase transition). Let a > 1 and V : Σ → R be a continuous function satisfying

V (x) = 1 n a + o( 1 n a ) if d(x, K) = 2 -n .
Then, for every γ 0, there exists a unique equilibrium state associated to -γV and it gives positive mass to every open set. The pressure function γ → P(γ) is analytic and positive on [0, ∞), although it converges to zero as γ → ∞.

Theorem 4 (Phase transition). Let a ∈ (0, 1) and V : Σ → R be a continuous function satisfying

V (x) = 1 n a + o( 1 n a ) if d(x, K) = 2 -n .
Then there exists γ 1 such that for every γ > γ 1 the unique equilibrium state for -γV is the unique invariant measure µ K supported on K. For γ < γ 1 , there exists a unique equilibrium state associated to -γV and it gives positive mass to every open set in Σ. The pressure function γ → P(γ) is positive and analytic on [0, γ 1 ).

These results show that case a = 1 (i.e., the Hofbauer potential) is the border between the regimes with and without phase transition. Whether there is a phase transition for the case a = 1 (i.e., the fixed point Ṽ ) or in other words the analog of the Hofbauer potential, discussed in [START_REF] Baraviera | The potential point of view for renormalization[END_REF], is much more subtle. We intend to come back to this question in a later paper.

The full shift (Σ, σ) can be interpreted geometrically by a degree 2 covering of the circle. The Manneville-Pomeau map can be viewed this way; it is expanding except for a single (one-sided) indifferent fixed point. When dealing with the Thue-Morse shift, it is natural to look for a circle covering with an indifferent Cantor set.

Theorem 5. There exist C 1 maps f a : [0, 1] , semi-conjugate to the full 2-shift and expanding everywhere except on a Cantor set K, such that K is conjugate to K in Σ and if a ∈ (0, 1), then -γ log f a has an ultimate phase transition.

Another geometric realization of the Thue-Morse shift and the prototype of renormalizability in one-dimensional dynamics is the Feigenbaum map. This quadratic interval map f q-feig has zero entropy, but when complexified it has entropy log 2. Moreover, it is conjugate to another analytic degree 2 covering map on C, which we call f feig , that is fixed by the Feigenbaum renormalization operator

R feig f = Ψ -1 • f 2 • Ψ
where Ψ is linear f -dependent holomorphic contraction. Arguments from complex dynamics give that P(-γ log |f feig |) = 0 for all γ 2, see Proposition 23. Because h top (f feig ) = log 2 on its Julia set, the potential -γ 1 log |f feig | has a phase transition for some γ 1 ∈ (0, 2]. When lifted to symbolic space,log |f feig | produces an unbounded potential V feig which is fixed by R. We can find a potential V u , which is constant on

(σ • H) k (Σ) \ (σ • H) k+1 (Σ)
for each k such that V feig -V u ∞ < ∞ and analyze the thermodynamic properties of V u . Although P(-γ 1 V feig ) = 0 for some γ 1 2, it is surprising to see that the potential V u exhibits no phase transition. We emphasize here an important difference with the Manneville-Pomeau case, where both the potential -γ log |f M P | and its countably piecewise version, the Hofbauer potential, which is constant on cylinder sets (H M P ) k (Σ) \ (H M P ) k+1 (Σ) = [0 2k+1 1], undergo a phase transition.

Theorem 6 (No phase transition for unbounded fixed point V u ). The unbounded potential V u given by

V u (x) = α(k -1) for x ∈ (σ • H) k (Σ) \ (σ • H) k+1 (Σ)
is a fixed point of R for any α ∈ R. If α < 0, then for every γ 0, there exists a unique equilibrium state for -γV u . It gives positive mass to any open set in Σ. The pressure function γ → P(-γV u ) is analytic and positive for all γ ∈ [0, ∞).

The exact definition of the equilibrium state for this unbounded potential can be found in Subsection 3.5. 1.3. Outline of the paper. In Section 2 we prove Theorems 1 and 2. In the first subsection we recall some and prove other results on the Thue-Morse substitution and its associated attractor K.

In Section 3 we study the thermodynamic formalism and prove Theorems 4, 5 and 6. This section uses extensively the theory of local thermodynamic formalism defined in [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF] and developed in further works of the author. Finally, in the Appendix, we explain the relation between the Thue-Morse shift and the Feigenbaum map, and state and prove Proposition 23.

2. Renormalization in the Thue-Morse shift-space 2.1. General results on the Thue-Morse shift-space. Let σ : Σ → Σ be the full shift

on Σ = {0, 1} N . If x = x 0 x 1 x 2 x 4 • • • ∈ Σ, let [x 0 . . . x n-1
] denote the n-cylinder containing x, and let xi = 1x i be our notation for the opposite symbol.

Recall that ρ 0 and ρ 1 are the fixed points of the Thue-More substitution, and that K = orb σ (ρ 0 ) = orb σ (ρ 1 ) is a uniquely ergodic and zero-entropy subshift. We denote by µ K its invariant measure.

We give here some properties for the Thue-Morse sequence that can be found in [START_REF] Berstel | Combinatorics on Words: Christophel Words and Repetitions in Words[END_REF][START_REF] Brlek | Enumeration of factors in the Thue Morse word[END_REF][START_REF] Dekking | On the Thue-Morse measure[END_REF][START_REF] De Luca | Morse sequence and a problem in semi groups[END_REF].

(1) Left-special words (i.e., words w such that both 0w and 1w appear in K) are prefixes of H k (010) or of H k (101) for some k 0. (2) Right-special words (i.e., words w such that both w0 and w0 appear in K) are suffixes of H k (010) or of H k (101) for some k 0. (3) Bispecial words (i.e., words w such that both left and right-special) are precisely the words τ k := H k (0), τk := H k (1), τ k τk τ k = H k (010) and τk τ k τk = H k (101) for k 0. There are four ways in which a word w can be extended to awb, i.e., with a symbol both to the right and left. It is worth noting that for w = τ k or τk , all four ways indeed occur in K, while for w = τ k τk τ k or τk τ k τk only two extensions occur. (4) The Thue-Morse sequence has low word-complexity:

p(n) = 3 • 2 m + 4r if 0 r < 2 m-1 , 4 • 2 m + 2r if 2 m-1 r < 2 m ,
where n = 2m + r + 1. (5) The Thue-Morse shift is almost square-free in the sense that if w = w 1 . . . w n is some word, then ww can appear in K, but not www 1 . The nature of the Thue-Morse substitution is such that ρ 0 and ρ 1 are concatenations of the words τ k and τk . Appearances of τ k and τk can overlap, but not for too long compared to their lengths, as made clear in Corollary 4

The next lemma shows that almost-invertibility of σ on K implies some shadowing close to K.

Lemma 1. For x ∈ Σ with d(x, K) < 2 -5 , let y, y ∈ K be the closest points in K to x and σ(x) respectively. If y = σ(y), then y starts as τ k , τk , τ k τk τ k or τk τ k τk for some k 3.

Proof. As y = σ(y), there is another z ∈ K such that σ(z) = y and z 0 = y 0 = x 0 . Let d be maximal such that y 1 . . . y d-1 = z 1 . . . z d-1 , so y d = z d . This means that the word y 1 . . . y d-1 is bi-special, and according to property (3) has to coincide with τ k , τk , τ k τk τ k or τk τ k τk for some k 3.

Due to the Cantor structure of K, the distance of an orbit to K is not a monotone function in the time. This is the main problem we will have to deal with.

Definition 2. Let x ∈ Σ be such that d(σ(x), K) < 2d(x, K). Then we say that we have an accident at σ(x). By extension, if d(σ k+1 (x), K) = 2d(σ k (x), K) for every k < n -1, but d(σ n (x), K) < 2d(σ n-1 (x), K), then we say that we have an accident at time n.

Proposition 3. Assume thatlog 2 d(x, K) = d and that b d is the first accident for the piece of orbit x, . . . , σ d (x), then

• x b x b+1 . . . x d-1 is a bispecial word for K; • d -b = 3 ε • 2 k for some k and ε ∈ {0, 1}; • x 0 . . . x d-1 is neither right-special nor left-special; • b 2 k if d -b = 2 k , 2 k+1 if d -b = 3 • 2 k .
Proof. Let y and y ∈ K be such that x and y coincide for d digits and σ b (x) and y coincide for at least db digits. Then

x b x b+1 . . . x d-1 = y b y b+1 . . . y d-1 = y 0 y 1 . . . y d-b-1
is a right-special because it can be continued both as y and y. The word x b x b+1 . . . x d-1 is also a left-special, because otherwise, by Lemma 1, only one preimage of y by σ would be in K and this would coincide with the word y b-1 y b . . . y d-b-1 . Then b -1 rather than b would be the first accident. By property (3) above, db = 3 ε 2 k . On the other hand, x 0 . . . x d-1 cannot be right-special, because otherwise there would be a point

x = x 0 . . . x d-1 xd • • • ∈ K with d(x, x) < 2 -d . If x 0 . . . x d-1 is left-special, then
To finish the proof of the proposition we need to check that the next accident cannot happen too early. Assume that x 0 . . .

x d-2 start as ρ 0 = r 0 r 1 r 3 . . . (the argument for ρ 1 is the same). Let π(n) = #{0 i < n : r i = 1} -#{0 i < n : r i = 0} count the surplus of 1's within the first n entries of ρ 1 . Clearly π(n) = 0 for even n and π(n) = ±1 otherwise. Assume the word τ k starts in ρ 0 at some digit m < 2 k . If π(m) = 1, then π(m + 3) = 2 while if π(m) = -1, then π(m + 7) = -2.
A similar argument works if τk stars at digit m. This shows that if τ k or τk can only start in ρ 0 at even digits. This means that we can take the inverse H -1 and find that τ k-1 (or τk ) start at digit m/2 < 2 k-2 in ρ 0 . Repeating this argument, we arrive at τ 3 or τ3 starting before digit 8 = 2 4-1 of ρ 0 , which is definitely false, as we can see by inspecting ρ 0 = 0110 1001 1001 0110 . . . . Note also that the bound 2 k is sharp, because τk starts in ρ 0 at entry 2 k . Finally, we need to answer the same question for bispecial words τ k τk τ k = τ k+1 τ k and τk τ k τk = τk+1 τk . The previous argument shows that neither can start before digit 2 k+1 , and also this bound is sharp, because τk τ k τk starts in ρ 0 at entry 2 k+1 . Corollary 4. Occurrences of τ k and τk cannot overlap for more than 2 k-1 digits.

Proof. We consider the prefix τ k of ρ 0 only, as the other case is symmetric. If the overlap was more than 2 k-1 digits, then τ k-1 or τk-1 would appear in ρ 0 before digit 2 k-1 , which contradicts part (3) of Proposition 3 Lemma 5. For each k 1, the Thue-Morse substitution

H satisfies K = 2 k -1 j=0 σ j • H k (K), where indicates disjoint union, so σ i • H k (K) ∩ σ j • H k (K) = ∅ for all 0 i < j < 2 k . Proof. Take x ∈ K, so there is a sequence (n k ) k∈N such that x = lim k σ n k (ρ 0 ). If this sequence contains infinitely many even integers, then x = lim k σ 2m k (ρ 0 ) = lim k σ 2m k • H(ρ 0 ) = lim k H • σ m k (ρ 0 ) ∈ H(K). Otherwise, (n k ) k∈N contains infinitely many odd integers and x = lim k σ 1+2m k (ρ 0 ) = lim k σ•σ 2m k •H(ρ 0 ) = lim k σ•H •σ m k (ρ 0 ) ∈ σ•H(K). Therefore K ⊂ H(K) ∪ σ • H(K). Now if x = H(a) = σ • H(b) ∈ K, then x = a 0 ā0 a 1 ā1 a 2 ā2 . . . = b0 b 1 b1 b 2 b2 . . . , so b0 = a 0 = ā0 = b 1 = b1 = a 1 = ā1 = b 2 = b 2 = a 2 . Therefore x = 101010 . . . or 010101 . . . , but neither belongs to K. Now for the induction step, assume K = 2 k -1 j=0 σ j • H k (K). Then since H is one-to-one, K = 2 k -1 j=0 σ j • H k (H(K) σ • H(K)) =   2 k -1 j=0 σ j • H k+1 (K)     2 k-1 j=0 σ j • H k • σ • H(K))   =   2 k -1 j=0 σ j • H k+1 (H(K)     2 k -1 j=0 σ j+2 k • H k+1 (K))   = 2 k+1 -1 j=0 σ j • H k+1 (K).
(σ • H) 2k (x) = aH 2k (b) . . . and (σ • H) 2k+1 (x) = āH 2k+1 (b).
Therefore H n (b) converges to ρ b , for b = 0, 1.

2.2.

Continuous fixed points of R on K: Proof of Theorem 1. We recall that we have

R(V ) = V • σ • H + V • H. Therefore R 2 V = R(V • σ • H + V • H) = V • σ • H • σ • H + V • σ • H 2 + V • H • σ • H + V • H 2 = V • σ 3 • H 2 + V • σ 2 • H 2 + V • σ • H 2 + V • H 2 ,
and in general

R n V = S 2 n V • H n where (S k V )(x) = k-1 i=0 V • σ i (x)
is the k-th ergodic sum.

Lemma 7. If V ∈ L 1 (µ K ) is a fixed point of R, then K V dµ K = 0.
Proof. For any typical (w.r.t. Birkhoff's Ergodic Theorem) y ∈ K we get

V (y) = (R n V )(y) = 2 n -1 j=0 V • σ j • H n (y). Hence 1 2 n V (y) = 1 2 n 2 n -1 j=0 V • σ j • H n (y).
The left hand side tend to 0 as n → ∞ and the right hand side tends to K V dµ K . Lemma 8. Let W be any continuous fixed point for R (on K). Then, for j = 0, 1,

W (01ρ j ) + W (10ρ j ) = 0 and W (1ρ j ) = W (10ρ j ) + W (0ρ j ). Proof. Using the equality W (x) = (RW )(x) = W • H(x) + W • σ • H(x) we immediately get: W • (σ • H) n (x) = W • H • (σ • H) n (x) + W • (σ • H) n+1 (x).
Using Lemma 6 on this new equality, we obtain

W (iρ j ) = W (i īρ j ) + W ( īρ j ),
for i, j ∈ {0, 1}. This gives the second equality of the lemma (for i = 1). The symmetric formula is obtained from the case i = 0, and then adding both formulas yields W (01ρ j ) + W (10ρ j ) = 0.

Remark 1. Lemma 8 still holds if the potential is only continuous at points of the form iρ j and i īρ j with i, j ∈ {0, 1}.

Recall the one-parameter family of potentials U c from [START_REF] Bowen | Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms[END_REF]. They are fixed points of R, not just on K, but globally on Σ. Let i : Σ → Σ be the involution changing digits 0 to 1 and vice versa. Clearly U c = -U c • i. We can now prove Theorem 1.

Proof of Theorem 1. Let W be a potential on K, that is fixed by R. We assume that the variations are summable:

∞ k=1 Var k (W ) < ∞.
We show that W is constant on 2-cylinders. Let x = x 0 x 1 . . . and y = y 0 y 1 . . . be in the same 2-cylinder (namely x 0 = y 0 and x 1 = y 1 ). Then, for every n, H n (x) and H n (y) coincide for (at least) 2 n+1 digits. Therefore [START_REF] Van Enter | Nonperiodic long-range order for fast-decaying interactions at positive temperatures[END_REF] . Again, the second equality in that lemma used for both ρ 0 and ρ 1 shows that W | [00] = W | [START_REF] Epstein | New proofs of the existence of the Feigenbaum functions[END_REF] = 0. Therefore W = U c with c = W (ρ 0 ), and the proof is finished.

|W (x) -W (y)| = |(R n W )(x) -(R n W )(y)| = |(S 2 n W )(H n (x)) -(S 2 n W )(H n (y))| 2 n+1 k=2 n +1 Var k (W ). Convergence of the series k Var k (W ) implies that 2 n+1 k=2 n +1 Var k (W ) → 0 as n → ∞. This yields that W is constant on 2-cylinders. Lemma 8 shows that W | [01] = -W |

2.3.

Global fixed points for R: Proof of Theorem 2. To give an idea why Theorem 2 holds, observe that the property

V (x) = 1 n + o( 1 n ) if d(x, K) = 2 -n
(so V vanishes on K but is positive elsewhere) is in spirit preserved under iterations of R, provided the shift σ doubles the distance from K. Let D denote the class of potentials satisfying this property. Choose x such that d(x, K) = 2 -m . Taking the limit of Riemann sums, and since R preserves the class of non-negative functions, we obtain

0 (R n V )(x) = 2 n -1 j=0 1 2 n m -j + 2 n -1 j=0 o( 1 2 n m -j ) → n→∞ (1 + o(1)) 1 0 1 m -t dt = (1 + o(1)) log m m -1 = 1 m + o( 1 m
).

However, it may happen that d(σ(y), K) < 2d(y, K) for some y = σ j • H n (x), in which case we speak of an accident (see Definition 2). The proof of the proposition includes an argument that accidents happen only infrequently, and far apart from each other.

Remark 2. We emphasize an important bi-product of the previous computation.

If V is of the form V (x) = o( 1 m ) when d(x, K) = 2 -m , then R n (V ) converges to 0. See also Proposition 9.
Proof of Theorem 2. The proof has three steps. In the first step we prove that the class D is invariant under R. In the second step we show that R n (V 0 ), with V 0 defined by

V 0 (x) = 1 m if d(x, K) = 2 -m
, is positive (outside K) and bounded from above. In the last step we deduce from the two first steps that there exists a unique fixed point and that it is continuous and positive. We also briefly explain why it gives the result for any V ∈ D.

Step 1. We recall that R is defined by (RV )(x

) := V • H(x) + V • σ • H(x). As H and σ are continuous, R(V ) is continuous if V is continuous. Let x ∈ Σ, then if x K ∈ K is such that (6) d(x, K) = d(x, x K ) = 2 -m , then d(H(x), H(x K )) = 2 -2m .
We claim that if m 3, then d(H(x), K) = d(H(x), H(x K )). Let us assume by contradiction that y ∈ K is such that d(H(x), K) = d(H(x), y) < d(H(x), H(x K )). By Lemma 5, y belongs either to H(K) or to σ • H(K). In the first case, say H(z) = y, we get

d(H(x), H(z)) < d(H(x), H(x K )).
This would yield d(x, z) < d(x, x K ) which contradicts the fact that d(x, K) = d(x, x K ).

In the other case, say y = σ • H(z), m 3 yields H(x) = a 0 ā0 a 1 ā1 a 2 ā2 . . . and σ • H(z) = b0 b 1 b1 b 2 b2 . . .. As in the proof of Lemma 5 this would show that y must start with 010101 or 101010. However, both are forbidden in K and this produces a contradiction. This finishes the proof of the claim.

Lemma 1 also shows that d(σ

• H(x), K) = 2 -(2m-1) = d(σ • H(x), σ • H(x K )). Therefore (7) (RV )(x) = V • H(x) + V • σ • H(x) = 1 2m + 1 2m -1 + o( 1 m ) = 1 m + o( 1 m ).
Step 2. We establish upper and lower bounds for R n (V 0 ) where V 0 is defined by

V 0 (x) = 1 m if d(x, K) = 2 -m . Let x ∈ Σ be such that d(x, K) = 2 -m
, and pick x K ∈ K such that x and x K coincide for exactly m initial digits. Due to the definition of K, m 2 (for any x) but we assume in the following that m 3. By [START_REF] Brlek | Enumeration of factors in the Thue Morse word[END_REF] we have d(

H n x, K) = d(H n x, H n x K ) = 2 -2 n m .
Assume that the first digit of x is 0. Then H n (x) coincides with ρ 0 at least for 2 n digits.

Assume now that H n x has an accident at the j-th shift, 1 j < 2 n , so there is

y ∈ K such that d(σ j • H n (x), y) < 2d(σ j • H n (x), σ j • H n (x K )). ρ 0 H n x d d j y ≈ 2 n H n x ρ 0 d d 2 n m H n x y H n x K Figure 2.
Half of the sum R n V can easily be estimated.

The last point in Proposition 3 shows j 2 n-1 . Therefore, using again that the sum approximates the Riemann integral,

(R n V 0 )(x) 1 2 n (2 n /2)-1 j=0 1 m -j/2 n → n→∞ 1 2 0 1 m -x dx 1 2m .
The worst case scenario for the upper bound is when there is no accident, and then

(8) (R n V 0 )(x) = 2 n -1 j=0 1 2 n m -j → n→∞ 1 0 1 m -x dx 1 m -1 as required.
Remark 3. Note that the largest distance between K and points

σ k (H n (x)) with k ∈ [[0, 2 n -1]] is smaller than 2 -(2 n m-2 n +1)
2 -2 n . This largest distance thus tends to 0 super-exponentially fast as n → ∞.

Step 3. We prove here equicontinuity for R n (V 0 ). Namely, there exists some positive κ, such that for every n, for every x and y

|R n (V 0 )(x) -R n (V 0 )(y)| κ | log 2 d(x, y)| , holds.
Assume that x and y ∈ Σ coincide for m digits. We consider two cases.

Case 1: d(x, K) = 2 -m =: d(x, z), with m < m (and z ∈ K). d d 2 n m H n (z) d d 2 n m H n (x) H n (y)
If there are no accidents for

σ j • H n (x) for j ∈ [[0, 2 n [[, then for every j, d(σ j (H n (x)), K) = d(σ j (H n (y)), K) = d(σ j (H n (x)), σ j (H n (z))),
and

V 0 (σ j (H n (x))) = V 0 (σ j (H n (y))). This yields (R n V 0 )(x) = (R n V 0 )(y).
Case 2: If there is an accident, say at time j 0 , then two sub-cases can happen.

Subcase 2-1. The accident is due to a point z that separates before 2 n m, see Figure 3.

d d j 0 2 n m H n (z) accident-j 0 H n (z) d d 2 n m H n (x)
H n (y) Again, we claim that V 0 (σ j (H n (x))) = V 0 (σ j (H n (y))) holds for j j 0 -1, but also for j j 0 but smaller than the (potential) second accident. Going further, we refer to cases 2-2 or 1.

Sub-case 2-2. The accident is due to a point much closer to H n (x) than to H n (y), see Figure 4.

In that case we recall that the first accident cannot happen before 2 n-1 , hence j 0 2 n-1 . Again, for j j 0 -1 we get V 0 (σ j (H n (x))) = V 0 (σ j (H n (y))). By definition of accident we get for j j 0 . This yields

max V 0 (σ j+2 n-1 (H n (x))) , V 0 (σ j+2 n-1 (H n (y))) 1 2 n m -2 n-1 -j d d j 0 2 n m H n (z) H n (z) d d 2 n m H n (x) accident-j 0 H n (y)
|(R n V 0 )(x) -(R n V 0 )(y)| 2 n-1 k=j 2 2 n m -2 n-1 -j = 1 2 n 2 n-1 k=j 2 m -1 2 -j 2 n
. This last sum is a Riemann sum and is thus (uniformly in n) comparable to the associated integral

1 2 0 1 m-1 2 -t dt 1 2(m-1) .
Step 4. Following Step 3, the family (

1 n n-1 k=0 R k (V 0 )) n is equicontinuous (and bounded),
hence there exists accumulation points. Let us prove that (

1 n n-1 k=0 R k (V 0 )) n actually con- verges.
Assume that V 1 and V 2 are two accumulation points. Note that both V 1 and V 2 are fixed points for R. They are continuous functions and Steps 1 and 2 show that they satisfy 1 2m

V i (x) 1 m + o( 1 m ), if d(x, K) = 2 -m . From this we get V 1 (x) -V 2 (x) 1 2m + o( 1 m ) = 1 2 V 0 (x) + o(V 0 (x)),
and then for every n,

V 1 -V 2 = 1 n n-1 k=0 R k ( V 1 ) -R k ( V 2 ) 1 2 1 n n-1 k=0 R k (V 0 ) + o(R k (V 0 )).
We recall from Remark 2 that o(R k (V 0 )) goes to 0 as k → ∞. Taking the limit on the right hand side along the subsequence which converges to V 2 we get

V 1 -V 2 1 2 V 2 + o(V 0 ), which is equivalent to 2 3 V 1 V 2 +o(V 0 ). Exchanging V 1 and V 2 we also get 2 3 V 2 V 1 +o(V 0 ). These two inequalities yield V 1 -V 2 1 3 V 0 + o(V 0 ) and V 2 -V 1 1 3 V 0 + o(V 0 ).
Again, applying R k on these inequalities and the Cesaro mean, we get

V 1 -V 2 1 3 V 2 + o(V 0 ) and V 2 -V 1 1 3 V 1 + o(V 0 ).
Iterating this process, we get that for every integer p, p p + 1

V 2 + o(V 0 ) V 1 p + 1 p V 2 + o(V 0 ). This proves V 1 -V 2 = o(V 0 ), i.e., ( V 1 -V 2 )(x) → 0 faster than V 0 (x) as x → K (see again Remark 2). But V 1 -V 2
is also fixed by R, so we can apply (8) with a factor o(V 0 ) in front. This shows that V 1 = V 2 , and hence the convergence of the Cesaro mean (

1 n n-1 k=0 R k (V 0 )) n .
This finishes the proof of Theorem 2.

2.4. More results on fixed points of R. The same proof also proves a more general result:

Proposition 9. Let a be a real positive number. Take

V (x) = 1 n a +o( 1 n a ) if d(x, K) = 2 -n . Then, for a > 1, lim n→∞ R n V ≡ 0 and for a < 1, lim n→∞ R n V ≡ ∞.
Proof. Immediate, since the Riemann sum as in (8) has a factor 2 n(1-a) in front of it.

Consequently, any V satisfying V (x) = 1 n + o( 1 n ) for d(x, K) = 2 -n belongs to the weak stable set V ∈ W s ( V ) of the fixed potential V from Theorem 2. However, W s ( V ) is in fact much larger: Proposition 10. If V (x) = 1 n g(x) for d(x, K) = 2 -n and g : Σ → R a continuous function, then 1 j j-1 k=0 R k (V ) → V • K g dµ K .
Proof. Take ε > 0 arbitrary, and take r ∈ N so large that sup |g|2 -r ε and if

d(x, K) = d(x, x K ) 2 -r , then |g(x) -g(x K )| ε. Next take k ∈ N so large that if k = r + s, then 1 2 s 2 s -1 i=0 g(σ i (y)) -g dµ K ε.
uniformly over y ∈ K. Then we can estimate

(R k V )(x) = 2 k -1 j=0 V • σ j • H k (x) 1 2 k 2 k -1 j=0 1 m -j 2 k g • σ j • H k (x) = 1 2 r 2 r -1 t=0 1 2 s 2 s -1 i=0 1 m -1 2 k (2 s t + i) g • σ 2 s t+i • H k (x) = 1 2 r 2 r -2 t=0 1 2 s 2 s -1 i=0 1 m -t 2 r + O(2 -r ) • K (g dµ K + O(ε)) + 1 2 r 1 2 s 2 s -1 j=0 1 m -1 2 k (2 k -2 s + j) sup |g| → 1 0 1 m -x dx • K g dµ K + O(3ε).
Since ε is arbitrary, we find lim sup k (R k V )(x)

1 m • K g dµ K + o( 1 m ). Similar to Step 2 in the proof of Theorem 2, we find lim sup k (R k V )(x) 1 2m • K g dµ K + o( 1 m
). From this, using the argument of Step 3 in the proof of Theorem 2, we conclude that for the Cesaro means, lim n

1 n n-1 k=0 (R k V )(x) = V (x) • K g dµ K .
2.5. Unbounded fixed points of R. The application to Feigenbaum maps discussed in the Appendix of this paper suggests the existence of unbounded fixed points V u of R as well. They can actually be constructed explicitly using the disjoint decomposition

Σ \ σ -1 {ρ 0 , ρ 1 } = k 0 (σ • H) k (Σ) \ (σ • H) k+1 (Σ) . If we set (9) V u | H(Σ) = g and V u (x) = V u (y) -V u • H(y) for x = σ • H(y), then V u is well-defined and RV u = V u on Σ \ σ -1 {ρ 0 , ρ 1 }. The simplest example is (10) V u | (σ•H) k (Σ)\(σ•H k+1 )(Σ) = (1 -k)α,
and we will explore this further for phase transitions in Section 3.

For x ∈ Σ \ σ • H(Σ) and x k = (σ • H) k (x), we have V u (x k ) = g(x) - k j=1 g • σ 2 j -2 • H j (x). Now for x ∈ [1] σ 2 j -2 • H j (x) → σ -2 (ρ 0 ) along odd j's, σ -2 (ρ 1 ) along even j's,
and the reverse formula holds for x ∈ [0]. In either case,

V (x k ) ∼ k 2 [g•σ -2 (ρ 0 )+g•σ -2 (ρ 1 )]. Therefore, unless g • σ -2 (ρ 0 ) + g • σ -2 (ρ 1 ) = 0, the potential V u is unbounded near lim k→∞ (σ • H) k (x) = {σ -1 (ρ 0 ) , σ -1 (ρ 1 )}, cf. Lemma 6.
Remark 4. A variation of this stems from the decomposition

Σ \ {ρ 0 , ρ 1 } = k 0 H k (Σ) \ H k+1 (Σ) .
In this case, if we define

V u | σ•H(Σ) = g and V u (x) = V u (y) -V u • σ(x) for x = H(y), then V u = RV u on Σ \ {ρ 0 , ρ 1 }.

Thermodynamic formalism

In this section we prove Theorems 4, 5 and 6. In the first subsection we define an induced transfer operator as in [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF] and use its properties. Then we prove both theorems.

3.1. General results and a key proposition. Let V : Σ → R be some potential function, and let J be any cylinder such that on it, the distance to K is constant, say δ J . Consider the first return map T : J → J, say with return time τ (x) = min{n 1 : σ n (x) ∈ J}, so T (x) = σ τ (x) (x). The sequence of successive return times is then denoted by τ k (x), k = 1, 2, . . . The transfer operator is defined as [START_REF] Epstein | New proofs of the existence of the Feigenbaum functions[END_REF] (L z,γ g)(x) = T (y)=x e Φz,γ (y) g(y)

where Φ z,γ (y

) := -γ(S n V )(y) -nz if τ (y) = n.
For a given test function g and a point x ∈ J, (L z,γ g)(x) is thus a power series in e -z .

These operators extend the usual transfer operator. They were introduced in [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF] and allow us to define local equilibrium states, i.e., equilibrium states for the potentials of the form Φ z,γ and the dynamical system (J, T ). These local equilibrium states are later denoted by ν z,γ .

We emphasize that, using induction on J, these operators L z,γ allow us to construct equilibrium states for potentials which do not necessarily satisfy the Bowen condition (such as e.g. the Hofbauer potential).

Nevertheless, we need the following local Bowen condition: there exists C V (possibly depending on J) such that ( 12)

|(S n V )(x) -(S n V )(y)| C V ,
whenever x, y ∈ J coincide for n := τ k (x) = τ k (y) indices. This holds, e.g. if V (x) depends only on the distance between x and K.

Lemma 11. Let x ∈ J and let γ and z be such that (L z,γ 1 I J )(x) < ∞. Then (L z,γ g)(y) < ∞ for every y ∈ J and for every continuous function g : J → R.

Proof. Note that for any x, y ∈ J, (L z,γ 1I J )(x) ≈ e ±C V (L z,γ 1I J )(y). Indeed, if x and y are two preimages of x and y in J, with the same return time n and such that for every k ∈ [[0, n]] σ k (x ) and σ k (y ) are in the same cylinder, then

|(S n V )(x ) -(S n V )(y )| C V .
Recall that J is compact, and that every continuous function g on J is bounded. Hence convergence (i.e., as power series) of (L z,γ 1I J )(x) ensures uniform convergence over y ∈ J for any continuous g. This finishes the proof of the lemma.

For fixed γ, there is a critical z c such that (L z,γ 1I J )(x) converges for all z > z c and z c is the smallest real number with this property. Lemma 11 shows that z c is independent of x. The next result is straightforward.

Lemma 12. The spectral radius λ z,γ, of L z,γ is decreasing in both γ and z.

We are interested in the critical z c and the pressure P(γ), both as function of γ. Both curves are decreasing (or at least non-increasing). If the curve γ → z c (γ) avoids the horizontal axis, then there is no phase transition: Proposition 13. Let V be continuous and satisfying the local Bowen condition (12) for every cylinder J disjoint and at constant distance from K. Then the following hold:

1. For every γ 0, the critical z c (γ) P(γ).

2. Assume that the pressure P(γ) > -γ V dµ K . Then there exists a unique equilibrium state for -γV and it gives a positive mass to every open set in Σ. Moreover z c (γ) < P(γ) and P(γ) is analytic on the largest open interval where the assumption holds. 3 If (L z,γ 1 I J )(ξ) diverges for every (or some) ξ and for z = z c (γ), then P(γ) > z c (γ) and there is a unique equilibrium state for -γV .

Proof. There necessarily exists an equilibrium state for -γV . Indeed, the potential is continuous and the metric entropy is upper semi-continuous. Therefore any accumulation point as ε → 0 of a family of measures ν ε satisfying

h νε (σ) -γ V dν ε P(γ)
is an equilibrium state.

The main argument in the study of local equilibrium states as in [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF] is that z > z c (γ) (to make the transfer operator "converges") and that V satisfies the local Bowen property [START_REF] Fisher | On discontinuity of the pressure[END_REF]. This property is used in several places and in particular, it yields for every x and y in J and for every n:

e -γC V (L n zγ 1I J )(x) (L n zγ 1I J )(y) e γC V .
To prove part 1., recall that

(L z,γ 1I J )(x) := ∞ n=1   x ,T (x )=x,τ (x)=n e -γ(SnV )(x )
  e -nz , which yields that z c = lim sup n 1 n log x ,T (x )=x,τ (x)=n e -γ(SnV )(x ) . To prove the inequality z c (γ) P(γ), we copy the proof of Proposition 3.10 in [START_REF] Leplaideur | On the t-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes[END_REF]. Define the measure ν as follows: for x in J and for each T -preimage y of x there exists a unique τ (y)-periodic point ξ(y) ∈ J, coinciding with y until τ (y). Next we define the measure ν n as the probability measure proportional to ξ(y),τ (y)=n e Φ P(γ),γ (ξ(y))

  n-1 j=0 δ σ j ξ(y)   = ξ(y),τ (y)=n e -γ(SnV )(ξ(y))-nP(γ)   n-1 j=0 δ σ j ξ(y)   .
The measure ν is an accumulation point of ( ν n ) n∈N . It follows from the proof of [18, Lemma 20.2.3, page 264] that [START_REF] Gallivotti | Ising model and Bernoulli schemes in one dimension[END_REF] z c (γ) h ν (σ)γ V d ν P(γ).

Remark 5. We emphasize that ν n (J) = 1 n for each n, which shows that ν(J) = 0. Now we prove part 2. Let µ γ be an ergodic equilibrium state for -γV . The assumption P(γ) > -γ V dµ K means that the unique shift-invariant measure on K cannot be an equilibrium state (since σ| K has zero entropy). Hence µ γ gives positive mass to some cylinder J in K c . Thus the conditional measure

(14) ν γ (•) := µ γ (• ∩ J)/µ γ (J).
is T -invariant (using the above notations).

We now focus on the convergence (as power series) of (L z,γ 1I J )(x) for any x ∈ J and z = P(γ). The inequality z c (γ) P(γ) does not ensure convergence of (L z,γ 1I J )(x) for z = P(γ). Again, we copy and adapt arguments from [START_REF] Leplaideur | On the t-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes[END_REF]Proposition 3.10] to get that (L z,γ 1I J )(x) converges and that the Φ z,γ -pressure is non-positive for z = P(γ).

In the case z > P(γ), so z > z c (γ), we can apply the local thermodynamic formalism for Φ z,γ . Moreover z > z c (γ) means that ∂ ∂z (L z,γ 1I J )(x) converges. This implies by [19, Proposition 6.8] that there exists a unique equilibrium state ν z,γ on J for T and for the potential Φ z,γ , and that the expectation J τ dν z,γ < ∞. In other words, there exists a shift-invariant probability measure µ z,γ such that µ z,γ (J) > 0, and ν z,γ (•) := µ z,γ (• ∩ J) µ z,γ (J) .

The equality h νz,γ (T ) + Φ z,γ dν z,γ = log λ z,γ (the spectral radius for L z,γ ) shows that

h µz,γ (σ) -γ V dµ z,γ = z + µ z,γ (J) log λ z,γ .
As z > P(γ) we get λ z,γ 1. Now the Bowen property of the potential shows that for every x ∈ J and for every n 1:

(L n z,γ 1I J )(x) = e γC V λ n z,γ .
The power series is decreasing in z, thus the monotone Lebesgue convergence theorem shows that it converges for z = P(γ). For this value of the parameter z, the spectral radius λ P(γ),γ 1. Following [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF], there exists a unique local equilibrium state, ν P(γ),γ with pressure log λ P(γ),γ 1. This proves that the Φ P(γ),γ -pressure is non-positive. Now, we prove that the Φ z,γ -pressure is non-negative for z = P(γ). Indeed, by Abramov's formula

0 = h µγ (σ) -γ V dµ γ -P(γ) = µ γ (J) h νγ (T ) -γ (S τ (x) V )(x) dν γ (x) -P(γ) τ dν γ , which yields h νγ (T ) -γ S τ (x) (V )(x) -P(γ)τ (x) dν γ (x) = 0.
Finally, as the Φ P(γ),γ -pressure is non-negative and non-positive, it is equal to 0. It also has a unique equilibrium state which is a Gibbs measure (in J and for the first-return map T ). As the conditional measure ν γ has zero Φ P(γ),γ -pressure, it is the unique local equilibrium state.

The local Gibbs property proves that ν γ gives positive mass to every open set in J, and the mixing property shows that the global shift-invariant measure µ γ gives positive mass to every open set in Σ. We can thus copy the argument to show it is uniquely determined on each cylinder which does not intersect K (here we use the assumption that the potential satisfies [START_REF] Fisher | On discontinuity of the pressure[END_REF] for each cylinder J with empty intersection with K).

It now remains to prove analyticity of the pressure. Equality [START_REF] Gallivotti | Ising model and Bernoulli schemes in one dimension[END_REF] gives z c (γ) h ν (σ)γ V d ν. Remark 5 states that ν(J) = 0, and uniqueness of the equilibrium state shows that ν cannot be this equilibrium state (otherwise we would have ν(J) > 0). Hence, z c (γ) is strictly less than P(γ). Then, we use [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] to get analyticity in each variable z and γ, and the analytic version of the implicit function theorem (see [START_REF] Range | Holomorphic Functions and Integral Representations on Several Complex Variables[END_REF]) shows that P(γ) is analytic.

The proof of part 3 is easier. The divergence of L z,γ (1I J )(ξ) for some ξ and z = z c (γ) ensures the divergence for every ξ, and then Lemma 12 and the local Bowen condition show that λ z,γ goes to ∞ as z goes to z c (γ). This means that there exists a unique Z > c (γ) such that λ Z,γ = 1. Using the work done in the proof of point 2, we let the reader check that necessarily Z = P(γ) and the local equilibrium state produces a global equilibrium state (see also [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF]).

This finishes the proof of the proposition.

Actually, Proposition 13 says a little bit more. If the second assumption is satisfied, namely P(γ) > -V dµ K , then the unique equilibrium state for V in Σ is the measure obtained (using Equation ( 14)) from the unique equilibrium state ν P(γ),γ for the dynamical system (J, T ) and associated to the potential Φ P(γ),γ . Therefore, two special curves z as function of γ appear, see Figure 5. The first is z c (γ), and the second is P(γ), defined by the implicit equality log λ P(γ),γ = 0. We claim that these curves are convex.

z = P(γ) z c (γ) γ z Figure 5
. Two important values of z as function of γ.

Counting excursions close to

K. Let x ∈ Σ and n ∈ N be such that for k ∈ [[0, n -1]], d(σ k (x), K)
2 -5 δ J . We divide the piece of orbit x, σ(x), . . . , σ n-1 (x) into pieces between accidents. We take b 0 = 0 by default, and let y 0 ∈ K be the point closest to x. Inductively, set

b 1 = min{j 1 : d(σ j (x), K) d(σ j (x), σ j (y 0 ))}, y 1 ∈ K is point closest to σ b 1 (x). b 2 = min{j 1 : d(σ j+b 1 (x), K) d(σ j+b 1 (x), σ j (y 1 ))}, y 2 ∈ K is point closest to σ b 1 +b 2 (x). b 3 = min{j 1 : d(σ j+b 1 +b 2 (x), K) d(σ j+b 1 +b 2 (x), σ j (y 2 )}, y 3 ∈ K is point closest to σ b 1 +b 2 +b 3 (x) . . . . . . . . . and d j = -log 2 d(σ i<j b i (x), K) = -log 2 d(σ i<j b i (x), y j-1
) expresses how close the image of x is to K at the j -1-st accident.

Following Proposition 3, d j -b j is of the form 3 ε j 2 k j , with ε j ∈ {0, 1} and d j+1 > d j -b j by definition of an accident. One problem we will have to deal with, is to count the possible accidents during a very long piece of orbit: if we know d jb j can we determine the possible values of d j ? As it is stated in Subsection 2.1, accidents occur at bispecial words which have to be prefixes of τ n τn τ n or τn τ n τn , and are words of the form τ k , τk , τ k τk τ k or τk τ k τk .

From now on, we pick some non-negative potential V and assume it satisfies hypotheses of Proposition 13. Namely, our potentials are of the form V (x) = 1 n a +o( 1 n a ) if log 2 (d(x, K)) = -n. They satisfy hypotheses of Proposition 13, and furthermore, the Birkhoff sums are locally constant.

Moreover, for x and y in J coinciding until n = τ (x) = τ (y), the assumption d(K, J) = δ J = d(x, K) = d(y, K) shows that for every j n, d(σ j (x), K) = d(σ j (y), K) holds. Hence Φ .,γ satisfies the local Bowen property [START_REF] Fisher | On discontinuity of the pressure[END_REF].

Let x be a point in J. We want to estimate (L z,γ 1I J )(x). Let y be a preimage of x for T . To estimate Φ(y), we decompose the orbit y, σ(y), . . . , σ τ (y)-1 (y) where σ j (y) is reasonably far away from K (let c r 0 be the length of such piece) and excursions close to K. Definition 14. An excursions begins at ξ := σ s (y) when ξ starts as ρ 0 or ρ 1 for at least 5log 2 δ J digits (i.e., d(ξ , ρ 0 ) and d(ξ , ρ 0 ) δ J 2 -5 ) and ends at ξ := σ t (y) where t > s is the minimal integer such that d(ξ , K) > δ J 2 -5 .

If σ i (y) is very close to K, then due to minimality of (K, σ) it takes a uniformly bounded (from above) number of iterates for an excursion to begin.

Note that each cylinder for the return map T is characterized by a path

c 0 , b 1,1 , b 1,2 , . . . , b 1,N 1 first excursion , c 1 , b 2,1 , . . . , b 2,N 1 second excursion , c 2 , . . . , c M -1 , b M,1 , b M,2 , . . . , b M,N 1 M -th excursion , c M .
Any piece of orbit between two excursions or before the first excursion or after the last excursion is called a free path. Let s r and t r be the times where the r-th free path and r-th excursion begin. Since J is disjoint from K, each piece c r of free path takes at least two iterates, so c r 2 for 0 r M .

Due to the locally constant potential we are considering, (L z,γ 1I J )(x) is independent of the point x where it is evaluated. Hence, for the rest of the proofs in this section, unless it is necessary, we shall just write L z,γ 1I J . Our strategy is to glue on together paths in functions of their free-paths and the numbers of accidents during an excursion. This form clusters and the contribution of a cluster considering N accidents is of the form

(15) E z,γ (1I J ) := N 1 allowed (b j ) N j=1 , (d j ) N j=1 exp   -γ N j=1 S j V   exp   - N j=1 b j z   D N A N ,
where S j V is the Birkhoff sum of the potential after the j th accident but before the next one and the quantity D N = e ϕ N -(d N -b N )z is the contribution of the last part of the orbit after the N th accident. By definition of an accident, this contribution is larger than if there would be no accident. Therefore for non-negative z,

e -(d N -b N )z (d N -b N ) γ D N 1.
The quantity A N is the sum of the contribution of the cluster with N accident.

Thus we have

(L 0,γ 1I J )(x) =    c 0 5 free c 0 -paths e -γ c 0 -1 n=0 V (σ n (y))-c 0 z    ×    M 0       (cr) M r=1 free cr-paths e -γ cr -1 n=0 V (σ n+sr (y))-crz    × (E z,γ (1I J )) M       . ( 16
)
3.3. The potential n -a : Proofs of Theorems 3 and 4 .

Proof of Theorem 3.

Here we deal with the case a > 1 and

V (x) = n -a if d(x, K) = 2 -n . Proof of Theorem 3. Since a > 1, d n=d-b+1 1 n a d d-b 1 x a dx = 1 a -1 1 (d -b) a-1 - 1 d a-1 1 a -1 < ∞,
for all values of b < d. To find a lower bound for E z,γ (1I J ) in [START_REF] Georgii | Gibbs measures and phase transitions[END_REF], it suffices to take only excursions with a single accident, and sum over all possible

d 1 with d 1 -b 1 = 2 k . Then E z,γ (1I J ) k e -γ/(a-1) = ∞,
regardless of the value of γ > 0.

By Proposition 13 (part 1) we get P(γ) > z c (γ) 0 = -γ V dµ K . Then Proposition 13 (part 3) ensures that there is no phase transition and that γ → P(γ) is positive and analytic on [0, ∞).

To finish the proof of Theorem 3, we need to compute lim γ→∞ P(γ). Let µ γ be the unique equilibrium state for -γV . Then

P(γ) γ = h µγ γ -V dµ γ ,
which yields lim sup γ→∞ P(γ) γ 0, hence lim sup γ→∞ P(γ) 0. On the other hand, P(γ) 0 = h µ K -V dµ K , hence lim inf γ→∞ P(γ) 0.

3.3.2.

Proof of Theorem 4 for a special case. Now take a ∈ (0, 1) and

V (x) = n -a if d(x, K) = 2 -n , so Φ z,γ (x) = -γS n V (x) -nz = -γ n k=1 k -a -nz.
The potential is locally constant on sufficiently small cylinder sets. It thus satisfies the local Bowen condition [START_REF] Fisher | On discontinuity of the pressure[END_REF] and the hypotheses of Proposition 13 hold.

Recall that

d n=d-b+1 1 n a d d-b 1 x a dx = 1 1 -a d 1-a -(d -b) 1-a ,
and we shall replace the discrete sum by the integral. The error involved in this can be incorporated in the changed coefficient (1 ± ε)γ.

Our goal is to prove that z c (γ) = 0 (for every γ) and that L 0,γ (1I J )(x) → 0 as γ → ∞ (for any x ∈ J). This will prove that there is a phase transition.

Lemma 15. The series (L z,γ 1 I J )(ξ) diverges for z < 0.

Proof. We employ notations from [START_REF] Georgii | Gibbs measures and phase transitions[END_REF] with our new V . In the full shift all orbits appear, and we are counting here only orbits which have only one excursion close to K without accident. For each j, we consider a piece of orbit of length 2 k+1 (1 + 2j), coinciding with a piece of orbit within K, and then "going back" to J. The quantity E z,γ (1I J ) is larger than the contribution of these excursions, which is

A k 1 (z) ∞ j=1 e -γ 1-a ((2 k+1 (1+2j)) 1-a -1)-2 k+1 (1+2j)z .
As a < 1, -2jz is eventually larger than (2 k+1 (1 + 2j)) 1-a for z < 0 and the series trivially diverges. Then, E z,γ (1I J ) diverges as well, and [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] shows that L z,γ (1I J ) diverges for every initial point x ∈ J.

Let us now consider the case z = 0. As we are now looking for upper bounds, we can consider the b j 's and the d j 's as independent and sum over all possibilities (and thus forget the condition d j+1 > d jb j ). Note that we trivially have D N 1.

= 2 k ( j 2 -2). Let B(z) := ∞ k=4 ∞ j=1 e -γ 1-a (2 k (1+ j 2 )) 1-a -2 k(1-a) -j2 k-1 z . and C(z) := ∞ k=4 ∞ j=5 e -γ 1-a (2 k (1+ j 2 )) 1-a -3 1-a 2 k(1-a) -2 k-1 (j-4)z .
The quantity B(z) is an upper bound for the cluster with one excursion of the form db = 2 k , and C(z) is an upper bound for the cluster with one excursion of the form

d -b = 3 • 2 k .
Then multiplying N copies to estimate from above the contribution of excursion with N accidents we get

E z,γ (1I J ) N (B(z) + C(z)) N . Hence (L 0,γ 1I J )(x)    c 0 5 free c 0 -paths e -γ c 0 -1 n=0 V (σ n (y))-c 0 z    ×    M 0       (cr) M r=1 free cr-paths e -γ cr -1 n=0 V (σ n+sr (y))-crz    ×   N 1 (B(z) + C(z)) N   M       ,
where the sum over (c r ) M r=1 is 1 by convention if M = 0. The first factor (the sum over c 0 ) indicates the first cluster of free paths, and c 0 5 by our choice of the distance δ J .

Note that for the free pieces between excursions the orbit is relatively far from K, so there is ε > 0 depending only on δ J such that [START_REF] Hofbauer | Examples for the nonuniqueness of the equilibrium state[END_REF] -c r (γ + z)

cr-1 n=0 -γV (σ n+sr (y)) -c r z -c r (εγ + z).
An upper bound for L z,γ (1I J ) is obtained by taking an upper bound for B and C and majorizing the sum over the c r free paths by taking the sum over all the c and the upper bound in [START_REF] Hofbauer | Examples for the nonuniqueness of the equilibrium state[END_REF].

Proof of Theorem 4. Lemma 15 shows that for every γ, z c (γ) 0. Our goal is to prove that B(0) + C(0) can be made as small as wanted by choosing γ sufficiently large. This will imply that z c (γ) = 0 for sufficiently large γ and that the unique equilibrium state is µ K . We compute B(0) leaving the very similar computation for C(0) to the reader.

Apply the inequality 1 + u 1 + log(1 + u) for the value of u such that 1

+ u = (1 + j 2 ) 1-a , to obtain (1 + j 2 ) 1-a -1 log(1 + j 2 ) 1-a , whence e (1+ j
2 ) 1-a -1

(1 + j 2 ) 1-a . Raising this to the power -γ 1-a 2 k(1-a) and summing over j, we get

∞ j=1 e -γ 1-a 2 k(1-a) ((1+ j 2 ) 1-a -1) ∞ j=1 (1 + j 2 ) -γ2 k(1-a) 2 3 γ2 k(1-a) + ∞ 1 dx (1 + x 2 ) -γ2 k(1-a) = 1 + 3 γ2 k(1-a) -1 2 3 γ2 k(1-a) . Therefore B(0) = ∞ k=4 ∞ j=1 e γ 1-a 2 k(1-a) (1-(1+ j 2 ) 1-a ) ∞ k=4 1 + 3 γ2 k(1-a) -1 2 3 γ2 k(1-a)
is clearly finite and tends to zero as γ → ∞. Now to estimate (L z,γ 1I J )(x), we have to sum over the free periods as well and we have

(L 0,γ 1I J )(x) c 5 2 c e -εγc • M 0 c 1 2 c e -cεγ (E 0,γ 1I J )(ξ) M 32e -5εγ 1 -2e -εγ M 0   c 1 2 c e -cεγ N 1 (B(0) + C(0)) N   M . ( 18
)
The term in the brackets still tends to zero as γ → ∞, and hence is less than 1 for γ > γ 0 and some sufficiently large γ 0 . The double sum converges for such γ, so the critical z c (γ) 0 for γ γ 0 .

Lemma 15 shows that z c (γ) is always non-negative. Therefore z c (γ) = 0 for every γ γ 0 . In fact, for γ sufficiently large (and hence e -5εγ is sufficiently small), the bound ( 18) is less than one: for every x ∈ J, (L 0,γ 1I J )(x) < 1. This means that log λ 0,γ , i.e., the logarithm of the spectral radius of L z,γ , becomes zero at some value of γ, say γ 2 .

Lemma 12 says that the spectral radius decreases in z. On the other hand the pressure P(γ) is non-negative because V dµ K = 0. Moreover, the curve z = P(γ) is given by the implicit equality λ P(γ),γ = 1. Therefore, the curve γ → P(γ) is below the curve γ → log λ 0,γ . Thus it must intersect the horizontal axis at some γ 1 ∈ [γ 0 , γ 2 ] (see Figure 6).

For γ > γ 1 convexity yields P(γ) = 0, hence the function is not analytic at γ 1 and we have an ultimate phase transition (for γ = γ 1 ). Analyticity for γ < γ 1 follows from Proposition 13.

3.3.3.

Proof of Theorem 4 for the general case. Now we consider V such that

V (x) = n -a + o(n -a ) if d(x, K) = 2 -n . γ 1 z γ z c (γ) z = P(γ) z = log λ 0,γ γ 0 γ 2 Figure 6
. Phase transition at γ 1

For every fixed ε 0 , there exists some N 0 such that for every n N 0 and for x such that

d(x, K) = 2 -n , V (x) - 1 n a
ε 0 n a . We can incorporate this perturbation in the free path, assuming that any path with length less than N 0 is a free path. Then all the above computations are valid provided we replace γ by γ(1 ± ε 0 ). This does not affect the results.

W (x) = 1 n a + o( 1 n a ) if d(x, K) = 2 -n
, and it is continuous at dyadic points:

W (x 1 . . . x n 10 ∞ ) = W (x 1 . . . x n 01 ∞ ),
and is positive everywhere except on K where it is zero.

Proof. Let us consider the multi-valued function V • Π -1 on the interval. It is uniquely defined at each non-dyadic point. For a dyadic point, consider the two preimages x 1 . . . x n 10 ∞ and x 1 . . . x n 01 ∞ in Σ.

Case 1. The word x 1 . . . x n (which is K-admissible) has a single suffix in K, say 0. This means that x 1 . . . x n 0 is an admissible word for K but not x 1 . . . x n 1. Let x -:= x 1 . . . x n 01 ∞ and x + := x 1 . . . x n 10 ∞ . Then [START_REF] Leplaideur | Local product structure for equilibrium states[END_REF] d(x

+ , K) = 2 -n > d(x -, K) > 2 -n-4 ,
where the last inequality comes from the fact that x 1 . . . x n 0111 is not admissible for K.

We modify the potential V on the right side hand of the dyadic point Π(x 1 . . . x n 10 ∞ ) as indicated on Figure 7.

Π(x 1 . . . x n 01 ∞ ) Π(x 1 . . . x n 10 ∞ ) modification The inequalities of (19) yield

V (x -) = 1 (n + k) a = 1 n a - ak n a+1 + o( 1 n a+2 ) = V (x + ) + o(V (x + )),
where k is an integer in [START_REF] Avila | Hausdorff dimension and conformal measures of Feigenbaum Julia sets[END_REF][START_REF] Bowen | Symbolic dynamics for hyperbolic systems, Recent advances in topological dynamics[END_REF]. As the modification is done "convexly", the new potential W satisfies for these modified points

W (x) = 1 n a + o( 1 n a ) if d(x, K) = 2 -n .
Case 2. The word x 1 . . . x n (which is K-admissible) has two suffixes in K. It may be that x + and x -are at the same distance to K (see Figure 8). Then we do not need to change the potential around this dyadic point.

If V (x + ) = V (x -), neither x 1 . . . x n 0111 nor x 1 . . . x n 1000 are admissible for K and we modify the potential linearly in that region in the interval as Figure 9.

Again we have

V (x + ) = 1 (n + j) a = 1 n a + o( 1 n a ) and V (x -) = 1 (n + k) a = 1 n a + o( 1 n a ),
where j and k are different integers in {1, 2, 3, 4}. Hence, for these points too, W satisfies

W (x) = 1 n a + o( 1 n a ) if d(x, K) = 2 -n .
no modification Π(x 1 . . . x n 10 ∞ ) Π(x 1 . . . x n 01 ∞ ) for any Borel set B on which σ is one-to-one. Since we have a phase transition at γ 1 , P(γ 1 ) = 0. Note also that W is positive everywhere except on K where it vanishes. Now consider the measure Π * (ν a ) and its distribution function

θ a (x) := ν a ([0 ∞ , Π(x))) = ν a ([0 ∞ , Π(x)]),
the last equality resulting from the fact that ν a is non-atomic. We emphasize that Π maps the lexicographic order in Σ to the usual order on the unit interval [0, 1]. This enables us to define intervals in Σ, for which we will use the same notation [x, y].

Let us now compute the derivative of f a define by

f a := θ a • Π • σ • Π -1 • θ -1 a at some point x ∈ [0, 1].
For h very small we define y and y h in [0, 1] such that Π * ν a ([0, y]) = x and Π * ν a ([0, y h ]) = x + h. Also define y and y h such that Π(y) = y and Π(y h ) = y h . Then we get

f a (x + h) -f a (x) h = ν a ([σ(y), σ(y h )]) ν a ([y, y h ]) = ν a (σ([y, y h ])) ν a ([y, y h ]) = 1 ν a ([y, y h ]) [y,y h ] e γ 1 W dν a → h→0 e γ 1 W (y) .
This computation is valid if Π -1 (y) is uniquely determined (namely y is not dyadic). If y h is dyadic for some h, then we choose for y h the one closest to y.

If y is dyadic, then the same can be done provided we change the preimage of y by Π depending on whether we compute left or right derivative. Nevertheless, the potential W is continuous at dyadic points, hence f a has left and right derivative at every dyadic points and they are equal.

We finally get f a (x) = e γ 1 W •Π -1 (x) (this make sense also for dyadic points) and then log f a (x) = γ 1 W • Π -1 (x). Therefore f a is C 1 and as W is positive away from K and zero on K, f a is expanding away from K := θ a • Π(K) and is indifferent on K. For t ∈ [0, ∞), the lifted potential for -t log f a is -tγ 1 W • Π -1 , which has an ultimate phase transition for t = 1 and a ∈ (0, 1).

The case a > 1. Computations are similar to the case a < 1, except that we have to add the pressure for γ 1 . The construction is the same, but the map f a satisfies :

f a (x) = e γ 1 W •Π -1 (x)+P(γ 1 ) .
This extra term is just a constant and then, the thermodynamic formalism for -t log f a is the same that the one for -tγ 1 W • Π -1 .

3.5. Unbounded potentials: Proof of Theorem 6. We know from Subsection 2.5 that R fixes the potential V u , defined in [START_REF] Van Enter | Breaking of periodicity at positive temperatures[END_REF]. In this section we set g ≡ α, which gives

V u = α(k -1) on (σ • H) k (Σ) \ (σ • H) k+1 (Σ).
For the thermodynamic properties of this potential, the interesting case is α < 0 (see the Introduction before the statement of Theorem 6 and the Appendix).

Lemma 17. Let α < 0. Then V u dµ Σ V u dµ K = 0 for every shift-invariant measure probability µ.

Note that 1I Σ\Oη V u is a continuous function. Thus, lim ε→0 1I Σ\Oη V u dν ε exists and is equal to 1I Σ\Oη V u dν = (1β) 1I Σ\Oη V u dµ. As η → 0, this quantity decreases and converges to (1β) V u dµ (here we use µ(K) = 0). Therefore, passing to the limit in [START_REF] De Luca | Morse sequence and a problem in semi groups[END_REF] first in ε and then in η we get

lim inf ε→0 V u dν ε (1 -β) V u dµ.
Proof of Proposition 18. We repeat the argument given in the proof of Proposition 13 and adapt it as in [START_REF] Leplaideur | On the t-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes[END_REF]. Let ν ε be a probability measure such that [START_REF] Makarov | On thermodynamics of rational maps. II. Non-recurrent maps[END_REF] h νεγ V u dν ε P(γ)ε, and let ν be any accumulation point of ν ε . As V u is discontinuous we cannot directly pass to the limit ε → 0 and claim that the integral of the limit measure is the limit of the integrals. However, we claim that V u is continuous everywhere but at the four points 0ρ 0 , 0ρ 1 , 1ρ 0 and 1ρ 1 (see and adapt the proof of Lemma 6). These points are in K and their orbits are dense in K. We thus have to consider two cases.

• ν(K) = 0. Then a standard argument in measure theory says that we do not see the discontinuity, and passing to the limit as ε → 0 in [START_REF] Makarov | On thermodynamics of rational maps. II. Non-recurrent maps[END_REF],

P(γ) h ν -γ V u dν P(γ),
which means that ν is an equilibrium state. • ν(K) > 0. In this case we can write ν = βµ K + (1β)µ, where µ is a σ-invariant probability satisfying µ(K) = 0 and β belongs to (0, 1]. Therefore

h ν = βh µ K + (1 -β)h µ = (1 -β)h µ . Lemma 19 yields (23) lim inf ε→0 V u dν ε (1 -β) V u dµ.
Hence, passing to the limit in Inequality [START_REF] Makarov | On thermodynamics of rational maps. II. Non-recurrent maps[END_REF], Inequality [START_REF] De Melo | One Dimensional Dynamics[END_REF] shows that

P(γ) (1 -β)h µ -γ(1 -β) V u dµ.
This last inequality is impossible if β < 1, by definition of the pressure. This yields that ν ε converges to µ K , and h νε converges to 0. Then [START_REF] De Melo | One Dimensional Dynamics[END_REF] shows that P(γ) 0.

On the other hand P(γ) 0 because the pressure is larger than the free energy for µ K , which is zero. Therefore µ K is an equilibrium state.

In order to use Proposition 13 we need to check that V u satisfies the hypotheses.

Lemma 20. For every cylinder J which does not intersect K, the potential V u satisfies the local Bowen property (12).

Proof. Actually, V u satisfies a stronger property: if x = x 0 x 1 . . . and y = y 0 y 1 . . . are in J (a fixed cylinder with J ∩ K = ∅), if n is their first return time in J, and if x k = y k for any 0 k < n, then (S n V u )(x) = (S n V u )(y).

Assume that J is a k-cylinder, and assume without loss of generality that n > k. The coordinates x j and y j coincide for 0 j < n, but since J is a k-cylinder, we actually have

x j = y j for 0 j n + k -1.

We recall that V u is constant on sets of the form (σ

• H) m (Σ) \ (σ • H) m+1 (K). Therefore, to compute V u (z) for z ∈ Σ we have to know which set (σ • H) m (Σ) \ (σ • H) m+1 (K) it belongs to. Lemma 6 shows that z = z 0 , z 1 , . . . belongs to (σ • H) m (Σ) \ (σ • H) m+1 (K) if and only if z 1 . . . z 2 m coincides with [ρ 0 ] 2 m or [ρ 1 ] 2 m
and m is the largest integer with this property.

Let us now study V u (σ j (x)) (and V u (σ j (y))) for j between 0 and n -1. We have to find the largest integer m such that z j+1 . . . z j+1+2 m coincides with [ρ 0 ] 2 m or [ρ 1 ] 2 m . As J does not intersect K, the word x n , . . . , x n+k-1 (which is also the word y n , . . . , y n+k-1 ) is not admissible for K. Therefore, the largest m such that z j+1 . . . z j+1+2 m coincides with

[ρ 0 ] 2 m or [ρ 1 ] 2 m satisfies 2 m n -j + k -1.
In other words, the integer m only depends on the digits where σ j (x) and σ j (y) coincide. Therefore V u (σ j (x)) = V u (σ j (y)).

Remark 7. An important consequence of Proposition 18 and Lemma 20 is that the conclusions of Proposition 13 hold. Although the potential V u is not continuous (and in fact undefined at σ -1 ({ρ 0 , ρ 1 })), it satisfies the local Bowen condition, so that the discontinuity is "invisible" for the first return map to J. Proposition 18 then implies the existence of an equilibrium state. Furthermore, the critical z c (γ) P(γ). By a similar argument as used in [20, Proposition 3.10] it can be checked that the conclusion of Lemma 19 holds despite the discontinuity of V u .

Lemma 21. Take α < 0 and consider the potential V u and some cylinder set J disjoint from K. The critical parameter for the convergence of (L z,γ 1 I J )(x) satisfies z c (γ) 2 -e -γα+2 +1 > 0 for every γ ∈ R and x ∈ J.

Proof. We now explore the thermodynamic formalism of the unbounded fixed point V u of R given by Equation 10. This potential is piecewise constant, and the value on cylinder sets intersecting K can be pictured schematically (with α = -1) as follows:

ρ 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 • • • ρ 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 • • • V u 1 0 1 -1 1 0 1 -2 1 0 1 -1 1 0 1 -3 • • •
Here, the third line indicates the value of V u at σ n (ρ j ) for n = 0, 1, 2, 3, . . . and j = 0, 1.

A single ergodic sum of length b = 2 k+1 -2 k-i (with α < 0 arbitrary again) for points x in the same cylinder as ρ 0 or ρ 1 is

(S b V u )(x) = 2 k+1 -2 k-i -1 j=0 V u (σ j (x)) = -α(1 + i).
Therefore, the contribution of a single excursionis

Φ z,γ (ξ) = N j=1 b j -1 k=0 γα(1 -i j ) -b j z
where i = i j is such that b j = 2 k+1 -2 k-i . The contribution to (L z,γ 1I J )(x) of one cluster of excursions then becomes E z,γ (ξ) N 1 A N , where (assuming that z 0)

A = allowed b 1 e γα(1+i)-bz = k 1 k-1 i=0 e γα(1+i)-(2 k+1 -2 k-i )z e γα k 1 k-1 i=0 e iγα-2 k+1 z = e γα k 1
1e γαk 1e γα e -2 k+1 z e γα k 1

e -2 k+1 z

Take an integer M e -γα+2 and z = 2 -(M +1) . Then taking only the M first terms of the above sum, we get the the entire sum is larger than e γα M e -2 M +1 z e γα e -γα+2 e -1 = e > 1.

Therefore, A > 1 and A N diverges. Hence, the critical z c (γ) 2 -e -γα+2 +1 > 0 for all γ > 0.

Proof of Theorem 6. It is just a consequence of Proposition 13 that a phase transition can only occur at the zero pressure. This never happens, hence the pressure is analytic on [0, ∞) and there is a unique equilibrium state for -γV u .

Appendix: The Thue-Morse subshift and the Feigenbaum map

The logistic Feigenbaum map f q-feig : I → I is conjugate to unimodal interval map f feig , which solves a renormalization equation ( 24)

f 2 feig • Ψ(x) = Ψ • f feig (x)
, for all x ∈ I, where Ψ is an affine contraction depending on f feig . Note that f feig is not a quadratic map, but it has a quadratic critical point c. See [START_REF] Epstein | New proofs of the existence of the Feigenbaum functions[END_REF] and [START_REF] De Melo | One Dimensional Dynamics[END_REF]Chapter VI] for an extensive survey.

As a result of (24), f feig is infinitely renormalizable of Feigenbaum type, i.e., there is a nested sequence M k of periodic cycles of 2 k -periodic intervals such that each component of M k contains two components of M k+1 . The intersection A := ∩ k 0 M k is a Cantor attractor on which f feig acts as a dyadic adding machine. The renormalization scaling Ψ : M k → M crit k+1 , where M crit k is the component of M k containing the critical point, and on each M crit k we have f 2 k+1 feig • Ψ = Ψ • f 2 k feig . Furthermore, A coincides with the critical ω-limit set ω(c) and it attracts every point in I except for countably many (pre-)periodic points of (eventual) period 2 k for some k 0. Hence f feig : I → I has zero entropy, and the only probability measures it preserves are Dirac measures on periodic orbits and a unique measure on A. This means that f feig : I → I is not very interesting from a thermodynamic point of view. However, we can extend f feig to a quadratic-like map on the complex domain, with a chaotic Julia set J supporting topological entropy log 2, and its dynamics is a finite-to-one quotient of the full two-shift (Σ, σ). Equation ( 24) still holds for the complexification f feig : U 0 → V 0 (a quadratic-like map, to be precise), where Ψ is a linear holomorphic contraction, and U 0 V 0 are open domains in C such that U 0 contains the unit interval. Renormalization in the complex domain thus means that M crit 1 extends to a disks U 1 V 1 and f 2 feig : U 1 → V 1 is a two-fold branched cover with branch-point c. The little Julia set J 1 = {z ∈ U 1 : f 2n feig (z) ∈ U 1 for all n 0} is a homeomorphic copy under Ψ of the entire Julia set J , but it should be noted that most points in U 1 eventually leave U 1 under iteration of f 2 feig : U 1 is not a periodic disk, only the real trace M crit 1 = U 1 ∩ R is 2-periodic. The same structure is found at all scales:

M crit k = U k ∩ R, U k V k and f 2 k
feig : U k → V k is a two-fold covering map with little Julia set J k := {z ∈ U 1 : f 2 k n feig (z) ∈ U 1 for all n 0} = Ψ(J k-1 ). To explain the connection between f feig : J → J and symbolic dynamics, we first observe that the kneading sequence ρ (i.e., the itinerary of the critical value f feig (c)) is the fixed point of a substitution H feig : 0 → 11, 1 → 10.

Let Σ feig = orb σ (ρ) be the corresponding shift space. If we quotient over the equivalence relation x ∼ y if x = y or x = w0ρ and y = w1ρ (or vice versa) for any finite and possibly empty word w, then Σ feig ∼ is homeomorphic to A, and the itinerary map i : A → Σ feig / ∼ conjugates f feig to the shift σ.

To make the connection with the Thue-Morse shift, observe that the sliding block code π : Σ → Σ defined by

π(x) k = 1 if x k = x k+1 , 0 if x k = x k+1 ,
is a continuous shift-commuting two-to-one covering map. The fact that it is two-to-one is easily seen because if x k = 1y k for all k, then π(x) = π(y). Surjectivity can also easily be proved; once the first digit of π -1 (z) is chosen, the following digits are all uniquely determined. It also transforms the Thue-Morse substitution H into H feig in the sense that Here pl is the parity-lexicographical order by which z < pl z if and only if there is a prefix w such that z = w0 . . . , z = w1 . . . and w contains an even number of 1s, z = w1 . . . , z = w0 . . . and w contains an odd number of 1s.

H feig • π = π • H.
On the level of itineraries, the substitution H feig plays the role of the conjugacy Ψ:

i • Ψ(x) = H feig • i(x) for all x ∈ [c 2 , c 1 ].
Also let l denote the usual lexicographical order. Since the regions U i are pairwise disjoint, the sum in the final line 1, so our choice of K gives that the above quantity is bounded by 1. Therefore the radius of convergence λ 0,γ 1. Taking the logarithm and using Abramov's formula, we find that the pressure P(-γφ) 0.

Date:

  Version of March 20, 2012. Part of this research was supported by a Scheme 3 (ref 2905) visitor grant of the London Mathematical Society and a visiting professorship at the University of Brest.

1. 2 .

 2 Statements of results. The Thue-Morse substitution (4) H := H TM : 0 → 01 1 → 10 has two fixed points ρ 1 = 1001 0110 1001 0110 01 . . . and ρ 0 = 0110 1001 0110 1001 10 . . .

Lemma 6 .

 6 Let x be in the cylinder [ab] with a, b ∈ {0, 1}. Then the accumulation point of (σ • H) k (x) are 0ρ b and 1ρ b . More precisely, the (σ • H) 2k (x) converges to aρ b and (σ • H) 2k+1 (x) converges to āρ b . Proof. By definition of H we get H(x) = aāH(b) . . . Hence σ • H(x) = āH(b) . . . By induction we get

Figure 3 .

 3 Figure 3. Comparing sequence when the accident occurs before separation.

Figure 4 .

 4 Figure 4. Comparing sequence when the accident occurs after separation.

For a piece of

  orbit of length d and with an accident at b, db = 2 k , the possible values of d's are among 2 k (1 + j 2 ), j 1, and then b = 2 k-1 j. If db = 3 • 2 k , then the possible values of d's are among 2 k (1 + j 2 ) with j 5 and then b

3. 4 .

 4 The proof of Theorem 5. As a direct application of Theorem 4, we can give a version of the Manneville-Pomeau map with a neutral Cantor set instead of a neutral fixed point.Proof of Theorem 5. Pick a > 0, and consider V and γ 1 as in Subsection 3.3 (only for a < 1). For a > 1 we pick any positive γ 1 . Define the canonical projection Π : Σ → [0, 1] by the dyadic expansion:Π(x 0 , x 1 , x 2 , . . .) = j x j 2 j+1 .It maps K to a Cantor subset of [0, 1]. Only dyadic points in [0, 1] have two preimages under Π, namely x 1 . . . x n 10 ∞ and x 1 . . . x n 01 ∞ have the same image.Lemma 16. There exists a potential W : Σ → R such that

Figure 7 .

 7 Figure 7. The modification for words with a single suffix

Figure 8 .Figure 9 .B e γ 1 W

 891 Figure 8. No modification with two different suffixes

  For the two Thue-Morse fixed points of H we obtain π(ρ 0 ) = π(ρ 0 ) = ρ = 10111010101110111011101010111010 . . .

Figure 10 summarizesiFigure 10 .

 1010 Figure 10 summarizes all this in a single commutative diagram. The Cantor set K factorizes over Σ feig and hence over the Cantor attractor A. The intermediate space L factorizes over the real core [c 2 , c 1 ] in the Julia set J and we can characterize its symbolic dynamics by means of a particular order relation. Namely, itineraries i(z) of z ∈ [c 2 , c 1 ] are exactly those sequences that satisfy σ(ρ) pl σ n • i(z) pl ρ for all n 0.

Lemma 22 .

 22 Let [0] and[START_REF] Avila | Hausdorff dimension and conformal measures of Feigenbaum Julia sets[END_REF] denote the one-cylinders of Σ. The map π : ([0], l ) → (Σ, pl ) is order preserving and π : ([1], l ) → (Σ, pl ) is order reversing.Proof. First we consider [0] and let w = 0 n , then w0 • • • < l w1 . . . and (25)π(w0 . . . ) = 0 n . . . pl 0 n-1 1 • • • = π(w1 . . . ).Now if we change the k-th digit in w (for k 2), then still w0 < l w1 and both the k-th and k -1-st digit of π(w . . . ) change. This does not affect the parity of 1s in π(w) and so[START_REF] Range | Holomorphic Functions and Integral Representations on Several Complex Variables[END_REF] remains valid. Repeating this argument, we obtain that π is order-preserving for all words w starting with 0. Now for the cylinder[START_REF] Avila | Hausdorff dimension and conformal measures of Feigenbaum Julia sets[END_REF] and w = 10 n-1 , we find w0 • • • < l w1 . . . and π(w0 . . . ) = 10 n-1 . . . pl 10 n-2 1 • • • = π(w1 . . . ). K/λ 0 . Therefore, for any x ∈ U , L 0,γ (1I J )(x) = i,∃x ∈U i F (x )=x |F (x )| -γ

In[START_REF] Baraviera | The potential point of view for renormalization[END_REF] it was proven that R n (V ) converges to the fixed point V ; here we only get convergence in the Cesaro sense.

Proof. As in Lemma 6, the set (σ • H) k (Σ) = σ 2 k -1 • H k ([00] [START_REF] Van Enter | Nonperiodic long-range order for fast-decaying interactions at positive temperatures[END_REF] [01] [START_REF] Epstein | New proofs of the existence of the Feigenbaum functions[END_REF]) consists of four 2 k + 1-cylinders containing the points 1ρ 0 , 0ρ 0 , 1ρ 1 and 1ρ 1 respectively, and they are mapped into the two 2 k -cylinders containing ρ 0 and ρ 1 . In other words, (σ • H) k (Σ) = σ -1 •H k (Σ), and by Lemma 5, its next 2 k shifts are pairwise disjoint. Therefore

, its µ-mass is at most 2 -k for any shift-invariant probability measure µ. Since V u is decreasing in k (for α < 0), we can minimize the integral V u dµ by putting as much mass on (σ • H) k (Σ) as possible, for each k. But this means that the µ-mass of (σ k+1) for each k, and hence µ = µ K .

Remark 6. As a by-product of our proof, µ (σ • H) k (Σ) 2 -k for any invariant probability µ and k 2.

For fixed α < 0, the integral V u dµ is non-negative and we define for γ 0

Proposition 18. For any γ 0 there exists an equilibrium state for -γV u .

To prove this proposition, we need a result on the accumulation value lim inf ε→0 V u dν ε if {ν ε } ε is a family of invariant probability measures.

Lemma 19. Let ν ε be a sequence of invariant probability measures converging to ν in the weak topology as ε → 0. Let us set ν := (1β)µ + βµ K , where µ is an invariant probability measure satisfying µ(K) = 0 and β ∈ [0, 1]. Then,

Let ν ε be an invariant probability measure. Following the same argument as in the proof of Lemma 17 and in particular Remark 6, we claim that

holds. Then we have

The same argument shows that π reverses order for all words w starting with 1.

This lemma shows that π -1 • i([c 2 , c 1 ]) consists of the sequence s such that for all n, σ(ρ 1 ) l σ n (s) l ρ 1 if σ n (s) starts with 1,

However, the class of sequence carries no shift-invariant measures of positive entropy, and the thermodynamic formalism reduces to finding measures that maximize the potential. The measure supported furthest away from K is the Dirac measure on 01 (with π(01) = 1).

Instead, if we look at the entire Julia set J , the combination of π and the quotient map do not decrease entropy, and the potentiallog |f feig | has thermodynamic interest for the complexified Feigenbaum map f feig : J → J . Since Ψ is affine, differentiating [START_REF] Chr | Boundary Behaviour of Conformal Maps, Grundlehren der mathematischen Wissenschaften 299[END_REF] and taking logarithms, we find that

), its size is exponentially small in k and hence there is some fixed α < 0 such that

As shown in Section 3.5, V u exhibits no phase transition.

The following proposition for complex analytic maps is stated in general terms, but proves the phase transition of Feigenbaum maps in particular.

Proposition 23. Let f : C → C be an n-covering map without parabolic periodic points such that the omega-limit set ω(Crit) of the critical set is nowhere dense in its Julia set J , and such that there is some c ∈ Crit such that f : ω(c) → ω(c) has zero entropy and Lyapunov exponent. Then for φ = log |f | and every γ > 2, P(-γφ) = 0.

Proof. As f has no parabolic points,

Obviously, all the invariant measures µ supported on ω(c) have h µγ log |f |dµ = 0, so P(-γφ) 0.

To prove the other inequality, we fix γ > 2 and for some f -invariant measure µ, we choose a neighborhood U intersecting J but bounded away from orb(Crit) such that µ(U ) > 0. We can choose diam(U ) so small compared to the distance d(orb(Crit), U ) that K γ-1 < λ γ-2 0 where K is the distortion constant in the Koebe Lemma, see [START_REF] Chr | Boundary Behaviour of Conformal Maps, Grundlehren der mathematischen Wissenschaften 299[END_REF]Theorem 1.3]. Since K → 1 as κ := diam(U )/d(orb(Crit), U ) → 0, we can satisfy the condition on K by choosing U small enough.

Let F : ∪ i U i → U be the first return map to U . Each branch F | U i = f τ i | U i , with first return time τ i > 0 can be extended holomorphically to f τ i : V i → f τ i (U i ) where f τ i (V i ) contains a disc around f τ i (U i ) of diameter d(orb(Crit), U ) diam(f τ i (U i ))/κ. Hence the Koebe Lemma implies that the distortion of f τ i | U i is bounded by K = K(κ). Furthermore, since each U i contains a τ i -periodic point of multiplier λ 0 , we have diam(U i )/diam(U )