Dynamical properties of certain continuous self maps of the Cantor set

Given a dynamical system (X,f) with X a compact metric space and a free ultrafilter p on N, we define for all x∈X. It was proved by A. Blass (1993) that x∈X is recurrent iff there is p∈N⁎=β(N)∖N such that fp(x)=x. This suggests to consider those points x∈X for which fp(x)=x for some p∈N⁎, which are called p-recurrent. We shall give an example of a recurrent point which is not p-recurrent for several p∈N⁎. Also, A. Blass proved that two points x,y∈X are proximal iff there is p∈N⁎ such that fp(x)=fp(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 σf:N{0,1}→N{0,1} by σf(x)(k)=x(f(k)) for every k∈N and for every x∈N{0,1} (the shift map on N{0,1} is obtained by the function k↦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(N{0,1},σf) is homeomorphic to β(N). Two functions g,h:N→N are defined so that E(N{0,1},σg) is homeomorphic to the Cantor set, and E(N{0,1},σh) is the one-point compactification of N with the discrete topology.