FROM LOCAL TO GLOBAL EQUILIBRIUM STATES: THERMODYNAMIC FORMALISM VIA AN INDUCING SCHEME

We present a method to construct equilibrium states via inducing. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Holder continuous potentials. It allows us to prove the existence of phase transition.

1. Settings 1.1. Goal. We consider a dynamical system (X, f ), where X is a compact metric space and f is topology mixing and local homeomorphism. For x in X, f −1 x is the inverse branch defined by x. It is a homeomorphism defined on a neighborhood of f (x) onto its image which is a neighborhood of x.
If φ : X → R is a continuous function or at least a Borel function, the pressure of φ is where h µ (f ) is the Kolmogorov entropy. The supremum is taken over the set of invariant probabilities. A measure which realizes the supremum is called an equilibrium state for φ. In the following it will be also referred to as a global equilibrium state. In the following, doing the thermodynamic formalism means study existence, uniqueness and other properties of global equilibrium states.
It is well-known that if (X, f ) is uniformly hyperbolic and φ is Hölder continuous, then there exists a unique equilibrium state (see e.g. [2,7]). In that case it is also a Gibbs state and has full support. The main heuristic explanation for this result is that hyperbolicity and Hölder regularity combine themselves and allow to construct the equilibrium state via the spectral elements of the transfer operator.
The existence of equilibrium states for less regular potentials and/or for systems with weaker hyperbolicity properties is a challenging task. In that case, the study of the spectral properties of the transfer operator is usually much harder. It turns out that, for several cases, one strategy is to consider induction.
Induction may also be a solution to deal with another question, related to the notion of phase transition. Beyond the question of existence of some equilibrium state, which can follows from abstract properties, one may want to get information on that equilibrium state as e.g. if it has or does not have full support. For this kind of question, induction can be a solution; it is actually a way to prove that an equilibrium state gives or does not give positive weight to some special set (see e.g. [3]).
Date: Version of August 27, 2013. AMSC : 37D35 Author wants to thank HB and JRC for readings and advises. 1 The goal of this note is to present a method based on induction to answer to these two previous questions. This method was actually presented in [9] for the case of uniformly hyperbolic dynamical system and Hölder continuous potential. It was then developed and extended in further works of the author; the note summarizes here all the principal steps using more general settings.
1.2. Markov set and induced system. We assume there exists some Markov set R: this is a proper set R =R such that for every n and for every x ∈R ∩ f −n (R), the set C n,f (x) := f −n x (R) is contained in R and f n (C n,f (x)) = R.
We consider the induced subsystems (R, F ) where F is the first return map: It is not necessarily well-defined everywhere, but it is however defined µ-a.e. for any invariant probability µ satisfying µ(R) > 0 (Poincaré's recurrence theorem). The main point is that the Markov property allows to well define the inverse branches: if x belongs toR and F (x) is well defined and also belongs toR, then we set C 1,F (x) := C τ (x),f (x). It is called the 1-cylinder of x (with respect to F ). The integer τ (x) is called the return-time of the 1-cylinder. By construction C 1,F (x) is a proper set, and the intersection of two different 1-cylinders has empty interior. They also form a partition of R, • up to points which never return to R by iterations of F , • and up to the fact that two 1-cylinders may have non-empty intersection (on their borders).
For x in the interior of a 1-cylinder C, τ (x) is well-defined and coincides with the return-time of the cylinder; this may not hold for point in ∂C. However we set F (x) = f n (x) if x belongs to ∂C and the return time for C is n. We point out that F may thus be multi-valued on these points, but these definitions coincide with the inverse branches. Therefore, the Markov property yields that for each x in R and for each 1-cylinder C, there exists a unique x ′ ∈ C such that F (x ′ ) = x. The set of x ′ 's is denoted as P re(x).
The main question we are interested in is : Is there a way to do the Thermodynamic formalism for (R, F ) and to recover (or to study the properties of) the equilibrium state for (X, f ) and φ ?
Then, the conditional measure µ := µ(. ∩ R) µ(R) is F -invariant. We remind the notation In By the Abramov formula (see [13] p. 257-258) we get This simple sequence of inequalities shows that the thermodynamic formalism for (X, f ) and φ is related to the thermodynamic formalism for (R, F ) and S τ (.) (φ)(.).
Our main assumptions on φ are that it is possible to study the thermodynamic formalism for Φ. Hypotheses are listed along the way. We first assume: (H1) Φ is continuous on each 1-cylinder. (H2) There exists C such that for x and y in the same 1-cylinder, We set for Z ∈ R, This is the transfer operator for (R, F ) and for the potential Φ − Z.τ (.).

Question 2.
For which Z can we do thermodynamic formalism ? Definition 1.1. Any equilibrium state for (R, F ) and for Φ − Z.τ (.) is called a local equilibrium state (associated to the parameter Z). It will be denoted by µ Z (if it exists).
Question 3. Among the measures µ Z , can we recover/find an equilibrium state for (X, f ) and φ ?
Roughly speaking, Question 3 means that we want to find Z such that the local equilibrium state say µ Z for Φ − Z.τ (.) is the induced measure of a global equilibrium state (for (X, f ) and φ). It is a reformulation of Question 1.

answers to questions
2.1. Local thermodynamic formalism.
Proposition 2.1. There exists a critical Z c ≥ −∞ such that −for every Z < Z c and for every x ∈ R, L Z (1 1 R )(x) = +∞, −and for every Z > Z c , for every ψ : R → R continuous and for every x ∈ R, L Z (ψ)(x) converges.
Theorem 1 (see [4], Lem. 3.4). If (X, f ) is a subshift of finite type and R is a cylinder and φ is continuous, then Z c is the pressure for φ of the set of points whose trajectory avoids R.
Proof. See Prop. 3.10 in [12] for a proof with a possible discontinuous potential.
From now on, we assume that φ has some regularity such that for every Z > Z c , L Z satisfies the hypothesis of the Ionescu-Tulcea & Marinescu theorem with some Banach spaces (V, V ) ⊂ C 0 (R): (i) if (ϕ n ) n∈N is a sequence of functions in V which converges in C 0 (R) to a function ϕ and if for all n ∈ N, ϕ n V ≤ C for some C > 0, then ϕ ∈ V and ϕ V ≤ C, (ii) L Z leaves V invariant and is bounded for (iv) there exists an integer n 0 and two constants 0 < a < 1 and 0 ≤ b < +∞ such that for all Under these hypotheses, L Z is quasi-compact on V: the spectrum is the union of finitely many isolated complex numbers which are eigenvalues with strictly dominating modulus and the essential spectrum contained in an open disk of radius the strictly smaller than the already mentioned isolated eigenvalues. Moreover, the spectral radius λ Z is a dominating eigenvalue (see [9]). The space V is "morally" the space of Hölder continuous functions. This is for instance the case in [9] or even in [12] despite the potential not being continuous 1 . The space of Hölder functions can also be used for the Hofbauer potential of the Manneville-Pomeau map (see [6,14] or Subsection 3.3).

Remark 1.
We emphasize that in the case that V is the set of Hölder continuous functions, hypothesis (iii) of Ionescu-Tulcea & Marinescu theorem is a direct consequence of (H2). Assuming Theorem 2 holds, the local equilibrium state µ Z is then of the form Continuity of H Z and positivity of L Z yield that H Z is strictly positive. The mixing hypothesis also shows that µ Z has full support in R.

2.2.
Local and global equilibria. The main question we are interested in is to know if among these µ Z , one could be the restriction of a/the gobal equilibrium state for (X, f ) and φ.
It is well known (see [5]) that there exists an f -invariant probability measure µ Z such that if and only if (2) τ dµ Z < +∞.

Now we have
Theorem 3. Inequality (2) holds if and only if for some x ∈ R L Z (τ )(x) converges. This is in particular the case if Z > Z c .
In this case the measure µ Z has full support (due to mixing) and satisfies Corollary 2.3. For every Z ≥ P, λ Z ≤ 1.
Proof. Equation (3) yields the desired inequality for Z > P ≥ Z c and continuity 2 in Z shows it also holds for Z = Z c One important point is that Z → λ Z is decreasing and analytic on ]Z c , +∞[ or even on .
It follows that Z → Z + µ Z (R) log λ Z attains its maximum either at the unique point Z where Then the main theorem is: equilibrium state µ such that µ(R) > 0, then (1) µ is the unique equilibrium state with full support, (2) The unique Z such that λ Z = 1 is Z = P, Remark 3. If (X, f ) is a subshift of finite type and φ is continuous, Theorem 1 and existence of the global equilibrium state show that the condition Z c < P yields the existence of some global equilibrium state giving positive weight to R. Consequently, there is uniqueness of the global equilibrium state and it is equal to µ P .

3.1.
For non-uniformly hyperbolic dynamics. In [12,11] the method is used for a horseshoe with homoclinic tangency. The potential is φ(x) := − log J u (x) := − log det Df |E u . It is noncontinuous due to the homoclinic tangency. Authors prove the existence and uniqueness of a global equilibrium state for β.φ and for every β ∈ R.

3.2.
For uniformly hyperbolic dynamics. In [3], the method is used to produce a freezing phase transition at positive temperature with ground state supported on a quasi-crystal. Namely, it is proved that there exists a continuous potential on {0, 1} N , say φ, and some β c > 0 such that the graphs of the pressure function for β.φ is strictly convex for β ∈ [0, β c ] and a half-line for β ≥ β c . Moreover, the exists a unique global equilibrium state for β.φ (may be except for β = β c ). It has full support for β < β c and is supported on some uniquely ergodic and zero-entropy (and different to a periodic orbit) for β > β c .
In [8] the method is used to construct a mixing system with an non-flat phase transition. It is also proved that after the transition, the system may have co-existence of several global equilibrium states despite the pressure function remains analytic.
In [10], the method is used to prove convergence at temperature zero of the global equilibrium state in a subshift of finite type and for a locally constant potential. Induction allows to control the different basins of different ergodic ground states, and to estimate how their relative weights vary in function of the inverse of the temperature β.

3.3.
Hofbauer potential or Manneville-Pomeau map. To finish with a simple example, we apply the method to the Hofbauer potential in This case is usually associated to the Manneville-Pomeau map, say e.g.
2x mod 1 if x ∈ [ 1 2 , 1]. One can find in [1] a description of why these two cases are associated, and actually similar.
For β > β c , no equilibrium state gives positive weight to [1], which means that δ 0 ∞ is the unique equilibrium state and the pressure is 0.
This last inequality shows that δ 0 ∞ cannot be an equilibrium state, hence, there exists an equilibrium state which gives positive weight to [1], and it is µ P(β) .