Piecewise Analytic Subactions for Analytic Dynamics

We consider a piecewise analytic expanding map f : [0, 1] → [0, 1] of degree d which preserves orientation, and an analytic positive potential g : [0, 1] → R. We address the analysis of the following problem: for a given analytic potential β log g, where β is a real constant, it is well known that there exists a real analytic (with a complex analytic extension to a small complex neighborhood of [0, 1]) eigenfunction φβ for the Ruelle operator. One can ask: what happen with the function φβ , when β goes to infinity? The domain of analyticity can change with β. The correct question should be: is 1 β log φβ analytic in the limit, when β → ∞? Under a uniqueness assumption, this limit, when β →∞, is in fact a calibrated subaction V (see bellow definition). We show here that under certain conditions and for a certain class of generic potentials this continuous function is piecewise analytic (but not analytic). In a few examples one can get that the subaction is analytic (we need at least to assume that the maximizing probability has support in a unique fixed point). The following question is related to the above problem. Denote m(log g) = max ν an invariant probability for f ∫ log g(x) dν(x), and μ∞ any probability which attains the maximum value. Any one of these probabilities μ∞ is called a maximizing probability for log g. We assume here that the maximizing probability is unique. The probability μ∞ is the limit of the Gibbs states μβ , for the potentials β log g. In this sense one case say that μ∞ corresponds to the Statistical Mechanics at temperature zero. In order to analyze ergodic properties of such probability μ∞, it is natural to associate to such f a bijective transformation σ̂, which acts on Σ̂ = Σ× [0, 1], where Σ = {1, 2, .., d}N. One can consider W the involution kernel associated to log g, where W : Σ̂ → R, and W (w, x) is defined for all w ∈ Σ and x ∈ [0, 1]. We show the existence of an analytic involution kernel for log g (in the sense that it is analytic in the second variable, for w fixed) and a interesting relation with the dual potential (log g)∗ defined in the Bernoulli space Σ. Using the above results we show that when μ∞ is unique, has support in a periodic orbit, the analytic function g is generic and satisfies the twist condition, then the calibrated sub-action V : [0, 1] → R for the potential log g is piecewise analytic. By definition, the calibrated subaction is the function V such that sup y such that f(y) = x {V (y) + log g(y) − m(log g) } = V (x). We assume the twist condition only in some of the proofs we present here. An interesting case where the theory can be applied is when log g(x) = − log f ′(x). In this case we relate the involution kernel to the so called scaling function. Date: April 22, 2009 (*) CIMAT, Guanajuato, Mexico (**) Instituto de Matemática, UFRGS, 91509-900 Porto Alegre, Brasil. Partially supported by CNPq, PRONEX – Sistemas Dinâmicos, Instituto do Milênio, and beneficiary of CAPES financial support, (***) Departamento de Matemática, ICMC-USP 13560-970 São Carlos, Brasil. Partially supported by CNPq 310964/2006-7 and FAPESP 2008/02841-4 .