The Relationship Between Entropy And Periodic Points

Let M be a compact manifold,f∈Diff1(M) be a C1diffeomorphism of M. In this paper, a generic property of Diff1(M) is given:there exists a residual subset W C Diff(M) such that every f∈W satisfies the following property: Here h(f) is the topological entropy of f. The main purpose of this paper is a gen-eralization of one of Katok’s theorem:in the paper [4] he introduced the relationship between the periodic points and the metric entropy of f in the case of that f is a C1+ε(ε>0) diffeomorphism.Moreover, I want to know the relationship between the periodic points and the metric entropy of flow, so this paper gives a new definition of metric entropy of flow; Let0be a continuous flow of a compact metric space X, m be aΦ invariant measure and hm(Φ)<∞. Then for m-almost every x∈X, we have (a)(b) hm(Φ,χ) is Φ-invariant;(c)∫X hm(Φ, χ)dm=hm(Φ).I hope this definition would be helpful for the study to the the relationship be-tween the periodic points and the metric entropy of flow.