Ultrafilter-limit points in metric dynamical systems

Given a free ultrafilter p on N and a space X, we say that x 2 X is the p-limit point of a sequence (xn)n2N in X (in symbols, x = p-limn!1 xn) if for every neighborhood V of x, {n 2 N : xn 2 V } 2 p. By using p-limit points from a suitable metric space, we characterize the selective ultrafilters on N and the P-points of N � = β(N) \ N. In this paper, we only consider dynamical systems (X, f), where X is a compact metric space. For a free ultrafilter p on N � , the function f p : X ! X is defined by f p (x) = p-limn!1 f n (x) for each x 2 X. These functions are not continuous in general. For a dynamical system (X, f), where X is a compact metric space, the following statements are shown: 1. If X is countable, p 2 Nis a P-point and f p is continuous at x 2 X, then there is A 2 p such that f q is continuous at x, for every q 2 A � .