Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents
The article deals with a connected compact Riemannian manifold \(M\), a \(C^2\) non-invertible map \(f: \;M \to M\) with no critical points, and an \(f\)-invariant probability measure \(\mu\). The author investigates sufficient and necessary conditions for the validity of equality \[ h_\mu(f) = F_\mu(f) - \int \sum_i \lambda_i^-(x) \, d\mu(x),\tag{1} \] where \(h_\mu(f)\) is the measure-theoretic entropy, \(\lambda_i^-(x)\) are all negative Lyapunov exponents at \(x\) (taking into account their multiplicity), and \(F_\mu(f)\) is the folding entropy, which can be defined as \[ F_\mu(f) =-\int \sum_{x\in f^{-1}(y)} p(x)\log p(x)\, d\mu(y), \] where \(p(x)\) is the conditional distribution of \(\mu\) on \(f^{-1}(y)\). For each \(x\), it is considered the stable manifold \(W^s(x)\), i.e., the set of all \(y\) whose orbits \(\{f^n(y)\}\) approach exponentially \(\{f^n(x)\}\). Locally one can define the conditional distributions of \(\mu\) on the stable manifolds. The main result is the following: equality (1) [Commun. Math. Phys. 240, No. 3, 531--538 (2003; Zbl 1078.37024)] holds true if and, under some additional condition, only if \(\mu\) has absolutely continuous conditional distributions on the stable manifolds. The additional condition mentioned above is the Hölder continuity of the Jacobian \(J_f\) computed with respect to \(\mu\). This paper continues previous author's investigations where he proved in a more general setting that \(h_\mu(f)\) does not exceed the right-hand side of (1) see [loc. cit.].
scientific article; zbMATH DE number 5530491