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

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} $$h_{\mu }(f)=F_{\mu }(f)-{\displaystyle \int {\sum }_{{\lambda }_i(x)<0}{\lambda }_i(x)d\mu (x),}$$ 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.