Some properties of positive entropy maps

Abstract We prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.) 60 (1) (1991), 11–14, Huang and Ye Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl. 117 (3) (2002), 259–272, Reddy The existence of expansive homeomorphisms on manifolds. Duke Math. J. 32 (1965), 627–632, Reddy and Robertson Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math. 17 (4) (1987), 673–681, Sindelarova A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268, and Zhou Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A 12 (suppl.) (1991), 22–27.