Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure
Let \(M\) be an \(n\)-dimensional compact \(C^{\infty}\) Riemannian manifold with \(n\geq 2\), and let \(f:M\rightarrow M\) be a \(C^r\)-diffeomorphism with \(1<r\leq \infty\). The paper is concerned with the metric entropy with respect to Sinai--Bowen--Ruelle measures of \(f\). The main result states that if \(f\) has a hyperbolic attractor \(\Lambda _f\), such that \(f\) is transitive on it, then there is a family of diffeomorphisms \(f_t\), \(0<t<1\), where \(f_1=f\) and the maps \(f\) and \(f_t\) can be chained by open balls on the \(C^1\)-topology. The maps \(f_t\) have hyperbolic attractors in such a way the restriction of \(f_t\) to such attractor is topologically conjugated to \(f|_{\Lambda _f}\) (and hence they have the same topological entropy on such sets), but the metric entropy of \(f_t\), relative to the unique SBR measure, converges to zero as \(t\) goes to zero.
scientific article; zbMATH DE number 5342979