Mathematics – Dynamical Systems
Scientific paper
2009-04-22
Mathematics
Dynamical Systems
This paper has been withdrawn by the authors. The present version has several results that are correct, but, there is a proble
Scientific paper
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 beta log g, where beta 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 phi_beta for the Ruelle operator. One can ask: what happen with the function phi_beta, when beta goes to infinity. The domain of analyticity can change with beta. The correct question should be: is 1/ beta log phi_beta analytic in the limit, when beta goes to infinity ? Under a uniqueness assumption, this limit, when beta goes to infinity, 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).
Contreras Gonzalo
Lopes Artur O.
Oliveira Elismar R.
Smania Daniel
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