DYNAMICAL SYSTEMS WITH SPECIFICATION

Abstract

Abstract. In this paper we prove that C 1 -generically, if a difieo-morphism f on a closed C 1 manifold M satisfles weak speciflcationon a locally maximal set ⁄ ‰ M then ⁄ is hyperbolic for f . As acorollary we obtain that C 1 -generically, every difieomorphism withweak speciflcation is Anosov. 1. IntroductionThe notion of the speciflcation was introduced by Bowen [3] to con-struct the equilibrium state and make a Markov partition of Axiom A difieomorphisms. After that the notion of weak speciflcation was intro-duced by Ruelle [10] to study variational principle for Z „ -action. Thenotions have turned out to be important in the study of ergodic theoryof dynamical systems (for more details, see [2, 3, 5, 7, 8, 10, 12, 13]).The deflnition of speciflcation (or weak speciflcation) seems to becomplicated and strong, but it is satisfled by many examples. Forexample, Lind [8] showed that hyperbolic toral automorphisms satisfyspeciflcation and ergodic central spin toral automorphisms satisfy weakspeciflcation. Moreover those properties for solenoidal automorphismsare well discussed by Aoki