Entropy production and folding of the phase space in chaotic dynamics

2011/04/12 by Eugen Mihailescu, Mihailescu, Eugen
Mathematics · Physics and Astronomy · #37A60 #37D35 #37D45 #82B05 #82C05 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.DS #math.MP #math.PR #msc:37A60 #msc:37D35 #msc:37D45 #msc:82B05 #msc:82C05

paper · pdf · doi:10.48550/arxiv.1104.2342

arxiv created 2011/04/12 · arxiv updated 2011/04/14

Abstract

We study the entropy production of Gibbs (equilibrium) measures for chaotic dynamical systems with folding of the phase space. The dynamical chaotic model is that generated by a hyperbolic non-invertible map f on a general basic (possibly fractal) set Λ; the non-invertibility creates new phenomena and techniques than in the diffeomorphism case. We prove a formula for the entropy production, involving an asymptotic logarithmic degree, with respect to the equilibrium measure μ_ϕ associated to the potential ϕ. This formula helps us calculate the entropy production of the measure of maximal entropy of f. Next for hyperbolic toral endomorphisms, we prove that all Gibbs states μ_ϕ have non-positive entropy production e_f(μ_ϕ). We study also the entropy production of the inverse Sinai-Ruelle-Bowen measure μ^- and show that for a large family of maps, it is strictly negative, while at the same time the entropy production of the respective (forward) Sinai-Ruelle-Bowen measure μ⁺ is strictly positive.

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