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 entropy formula of folding type \(h_{\mu }(f)= F_{\mu }(f)-\displaystyle \int \sum \limits _{\lambda _i(x)<0} \lambda _i(x)d\mu (x), \) 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.
