Entropy formula of folding type for $C^{1+\hat{I}\pm }$ maps

Abstract

In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula of folding type} \[h_μ(f)= F_μ(f)-\displaystyle\int\sum\nolimits_{λ_i(x)<0} λ_i(x)dμ(x), \] which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of $C^{1+\hat{I}\pm }$ maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that $\hat{I}¼$ is an inverse SRB measure if and only if the folding-type entropy formula holds and the Jacobian series is integrable. To overcome the degeneracy, we develop Pesin theory for general $C^{1+\hat{I}\pm }$ maps.

35 pages