On some coding and mixing properties for a class of chaotic systems
Let \(f:M\rightarrow M\) be a \(C^{2}\), non-invertible map defined on a smooth Riemannian manifold \(M\). A locally maximal set \(\Lambda \) is an invariant compact set which has a neighborhood \(U\subset M\) with \(\Lambda =\cap _{n\in \mathbb{Z}}f^{n}(U)\). A basic set for \(f\) is a locally maximal set \(\Lambda \) such that \(f\) is topologically mixing on \(\Lambda \). As the map is non-invertible on \(\Lambda \), \(\Lambda \) is sometimes called a folded basic set or a basic set with overlaps. From the abstract: ``We study certain ergodic properties of equilibrium measures of hyperbolic non-invertible maps \(f\) on basic sets with overlaps \(\Lambda \). We prove that if the equilibrium measure \(\mu _{\phi }\) of a Holder potential \(\phi \) is 1-sided Bernoulli, then \(f\) is expanding from the point of view of a pointwise section dimension of \(\mu _{\phi }\). If the measure of maximal entropy \(\mu _{0}\) is 1-sided Bernoulli, then \(f\) is shown to be distance expanding on \(\Lambda \); and if \(\mu _{\phi }\) is 1-sided Bernoulli for \(f\) expanding, then \(\mu _{\phi }\) must be measure of maximal entropy. These properties are very different from the case of hyperbolic diffeomorphisms.'' Other related properties are also proved.
equilibrium measures for hyperbolic non-invertible maps
scientific article; zbMATH DE number 6109329
scientific article