Dynamical fibre bundles. Entropy and dimension
The author studies a \(C^{1,t}\)-map \(\varphi\) on a subset \(X\) of a Banach space \(E\) leaving a Borel probability \(\mu\) on \(X\) invariant. The set \(X\) is assumed to have finite fractal dimension. The main theorem provides inequalities relating local entropy, local \(\alpha\)-entropy and the sum of positive Lyapunov exponents. Under the additional assumptions that \(X\) is compact and that the tangent maps of \(\varphi\) are uniformly asymptotically compact, these inequalities imply the following corollary: If the fractal dimension of \(\mu\) and the Lyapunov dimension of the tangent bundle coincide, then a local version of Pesin's formula holds, i.e. \(h_{\mu}(x,\varphi)=\sum_ id_ i(x)\lambda^ +_ i(x)\). One should observe that even in the finite-dimensional case this result has a new aspect: It is not necessary to know a priori that \(\varphi\) is invertible.
scientific article; zbMATH DE number 93900
scientific article