h−AVERAGE-SHADOWING PROPERTY AND CHAOS MEHDI FATEHI NIA DEPARTMENT OF MATHEMATICS, YAZD UNIVERSITY, 89195-741 YAZD, IRAN E-MAIL:FATEHINIAM@YAZD.AC.IR (Received: 30 April 2013, Accepted: 29 May 2013) Abstract. Let X be a compact metric space and f : X −→ X be a homeo- morphism map. In this paper we are going to introduce the h−average shad- owing property for discrete dynamical system (X, f ). It is shown that f has the h−average shadowing property if and only if f -1 has the h−average shadowing property. We also prove that if f has the h-average shadowing property, then every point x is chain recurrent. The relation between chaos and shadowing property in discrete dynamical system is considered. AMS Classification: 37C50, 37C15 Keywords: Shadowing; average shadowing property; chaotic dynamical systems; chain recurrent; pseudo orbit. 1. Introduction Let (X, d) be a metric space and f : X −→ X be a homeomorphism map. For every positive integer n, we define f n inductively by f n = fof n-1 and f 0 is the identity map on X. Also f -n =(f -1 ) n where f -1 is inverse of f. If x ∈ X then the orbit of x under f is the sequence x = {f n (x)} n∈Z . We refer (X, f ) as a discrete dynamical system. A subset A in X is said to be f −invariant if f (A) ⊆ A. The concept of shadowing was investigated by many authors, for example see JOURNAL OF DYNAMICAL SYSTEMS & GEOMETRIC THEORIES VOL. 11, NUMBER 1-2 (2013) 39-49. c TARU PUBLICATIONS 39