Entropy production and folding of the phase space in chaotic dynamics

TL;DR

This paper investigates entropy production in chaotic systems with phase space folding, deriving formulas for hyperbolic non-invertible maps and analyzing Gibbs states' entropy production, revealing non-positivity and negativity in various measures.

Contribution

It introduces a new formula for entropy production in non-invertible hyperbolic maps and analyzes the sign of entropy production for Gibbs states and SRB measures.

Findings

Entropy production formula involving asymptotic logarithmic degree.

Gibbs states for hyperbolic toral endomorphisms have non-positive entropy production.

Inverse SRB measure's entropy production is strictly negative, while forward SRB measure's is positive.

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 $\Lambda$; 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 $\mu_\phi$ associated to the potential $\phi$. 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 $\mu_\phi$ have \textit{non-positive entropy production} $e_f(\mu_\phi)$. We study also the entropy production of the \textit{inverse Sinai-Ruelle-Bowen measure} $\mu^-$ 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 $\mu^+$ is strictly positive.