Type: Article

Dynamical properties of certain continuous self maps of the Cantor set

García-Ferreira, S.;

Abstract:

Given a dynamical system (X, f) with X a compact metric space and a free ultrafilter p on N, we define f(p)(x) = p-lim(n ->infinity) f(n)(x) for all x is an element of X. It was proved by A. Blass (1993) that x is an element of X is recurrent iff there is p is an element of N* = beta(N) \ N such that f(p)(x) = x. This suggests to consider those points x is an element of X for which f(p)(x) = x for some p is an element of N*, which are called p-recurrent. We shall give an example of a recurrent point which is not p-recurrent for several p is an element of N*. Also, A. Blass proved that two points x, y is an element of X are proximal iff there is p is an element of N* such that f(p)(x) = f(p)(y) (in this case, we say that x and y are p-proximal). We study the properties of the p-proximal points of the following continuous self maps of the Cantor set: For an arbitrary function f : N -> N, we define sigma(f) : {0,1}(N) -> {0, 1}(N) by sigma(f)(x)(k) = x(f(k)) for every k is an element of N and for every x is an element of {0, 1}(N) (the shift map on {0.1}(N) is obtained by the function k bar right arrow k + 1). Let E(X) denote the Ellis semigroup of the dynamical system (X, f). We prove that if f : N -> N is a function with at least one infinite orbit, then E({0, 1}(N), sigma(f)) is homeomorphic to beta(N). Two functions g, h : N -> N are defined so that E({0, 1}(N), sigma(g)) is homeomorphic to the Cantor set, and E({0, 1}(N), sigma(h)) is the one-point compactification of N with the discrete topology