Entropy formula of folding type for $C^{1+\alpha}$ maps

TL;DR

This paper extends the entropy formula of folding type to all inverse SRB measures of $C^{1+eta}$ maps, including degenerate cases, by developing Pesin theory for such maps.

Contribution

It establishes the entropy formula for inverse SRB measures of $C^{1+eta}$ maps, including those with degeneracy, and develops Pesin theory for these maps.

Findings

Entropy formula of folding type holds for all inverse SRB measures.

Equivalence between SRB measures and the folding entropy formula with integrable Jacobian.

Development of Pesin theory for $C^{1+eta}$ maps with degeneracy.

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} \[h_{\mu}(f)= F_{\mu}(f)-\displaystyle\int\sum\nolimits_{\lambda_i(x)<0} \lambda_i(x)d\mu(x), \] which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of $C^{1+\alpha}$ maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that $\mu$ 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 $C^{1+\alpha}$ maps.