Piecewise Analytic Subactions for Analytic Dynamics
Abstract
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).
Keywords
Cite
@article{arxiv.0904.3516,
title = {Piecewise Analytic Subactions for Analytic Dynamics},
author = {Gonzalo Contreras and Artur O. Lopes and Elismar R. Oliveira and Daniel Smania},
journal= {arXiv preprint arXiv:0904.3516},
year = {2011}
}
Comments
This paper has been withdrawn by the authors. The present version has several results that are correct, but, there is a problem in the use of sections 7 and 8 to derive generic properties for the set of analytic potentials g. All sections before this are OK