Epsilon-Equicontinuous Points and Epsilon-Shadowable Points

Let f : X -> X be a continuous map on a compact metric space ( X, d) with no isolated points. We introduce the concept of epsilon-equicontinuous point in ( X, f), indeed a point x. X, is called epsilon-equicontinuous point, x is an element of Eq(epsilon)(f), if there exists delta > 0 such that diam(f(n)(B(x, delta))) <= epsilon for every n is an element of Z(+) We give a system (X, f) such that for every epsilon > 0, Eq(epsilon) (f) not equal (empty set) but f has no equicontinuous point. Next, we study its basic properties and compare some property of epsilon-equicontinuous points in (X, f) with those in two related dynamical systems, inverse limit space (lim <-( X, f), sigma(f)) and hyper space (2(X), 2(f)). It is known that a topologically transitive with equicontinuity points is uniformly rigid, we give an epsilon version of that result. We also introduce the concept epsilon-chain continuity points and study relation between epsilon-chain continuity points and two concepts epsilon-shadowable points and epsilon-equicontinuous points.