Entropy production and folding of the phase space in chaotic dynamics

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 $\mathrm{\Lambda }$; the non-invertibility creates new phenomena and techniques than in the diffeomorphism case. We prove a formula for the $\mathit{\text{entropy}}\mathit{\text{production}}$, involving an asymptotic logarithmic degree, with respect to the equilibrium measure ${\mu }_{\varphi }$ associated to the potential $\varphi$. 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 ${\mu }_{\varphi }$ have $\mathit{\text{non-positive}}\mathit{\text{entropy}}\mathit{\text{production}}$ $e_f({\mu }_{\varphi })$. 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 $\mathit{\text{strictly}}\mathit{\text{negative}}$, while at the same time the entropy production of the respective (forward) Sinai-Ruelle-Bowen measure ${\mu }^{+}$ is strictly positive.