Entropy production and folding of the phase space in chaotic dynamics

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.