2009/04/22 by Gonzalo Contreras, Contreras, Gonzalo, Artur O. Lopes +5 · 1 citation
Mathematics · Physics and Astronomy · #37A05 #37D20 #37D35 #37E05 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.CV #math.DS #msc:37A05 #msc:37D20 #msc:37D35 #msc:37E05
paper · pdf · doi:10.48550/arxiv.0904.3516
This paper has been withdrawn by the authors.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
openalex publication_date 2009/04/22 · arxiv created 2011/01/19 · arxiv updated 2011/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 phibeta for the Ruelle operator. One can ask: what happen with the function phibeta, when beta goes to infinity. The domain of analyticity can change with beta. The correct question should be: is 1/ beta log phibeta 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).