In this paper we introduce the notions of strict persistence and weakly strict persistence which are stronger than those of persistence and weak persistence, respectively, and study their relations with shadowing property. In particular, we show that the weakly strict persistence and the weak inverse shadowing property are locally generic in Z(M). 1. Introduction Let (M; d) be a compact metric space and let f : M ! M be a homeomorphism (a discrete dynamical system on M). A sequence fxngn2Z is called an orbit of f, denote by o(x; f), if for each n 2 Z, xn+1 = f(xn) and is called a -pseudo-orbit of f if d(f(xn); xn+1) ; 8n 2 Z: We denote the set of all homeomorphisms of M by Z(M). Introduce in Z(M) the complete metric d0(f; g) = maxfmaxx2Md(f(x); g(x)); maxx2Md(f

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