2012/05/25 by Artur O. Lopes, Lopes, Artur O., Elismar R. Oliveira +3
Mathematics · Physics and Astronomy · #37A05 #37A45 #37C30 #37C35 #37F15 #90B06 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Optimization and Control (math.OC) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1205.5758
openalex publication_date 2012/05/25 · openalex created_date 2022/09/27 · openalex updated_date 2026/07/28
We consider a piecewise analytic real expanding map f: [0,1]\→ [0,1] of\ndegree d which preserves orientation, and a real analytic positive potential\ng: [0,1] \→ \ℝ. We assume the map and the potential have a complex\nanalytic extension to a neighborhood of the interval in the complex plane. We\nalso assume \log g is well defined for this extension.\n It is known in Complex Dynamics that under the above hypothesis, for the\ngiven potential \β ,\log g, where \β is a real constant, there\nexists a real analytic eigenfunction \φ_\β defined on [0,1] (with a\ncomplex analytic extension) for the Ruelle operator of \β ,\log g.\n Under some assumptions we show that \(1)/(\β) , \log \φ_\β\nconverges and is a piecewise analytic calibrated subaction. Our theory can be\napplied when \log g(x)=-\log f'(x). In that case we relate the involution\nkernel to the so called scaling function.\n