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arXiv · 1104.2342

Entropy production and folding of the phase space in chaotic dynamics

Eugen Mihailescu
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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 \textit{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 \textit{non-positive entropy production} $e_f(μ_ϕ)$. We study also the entropy production of the \textit{inverse Sinai-Ruelle-Bowen measure} $μ^-$ and show that for a large family of maps, it is \textit{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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Eugen Mihailescu. 2011-04-12. Entropy production and folding of the phase space in chaotic dynamics. https://arxiv.org/abs/1104.2342

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