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 f p ( x ) = p - lim n → ∞ f n ( x ) 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 f p ( x ) = x . This suggests to consider those points x ∈ X for which f p ( 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 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: arbitrary function f : N → N , we define σ f : { 0 , 1 } N → { 0 , 1 } N by σ f ( x ) ( k ) = x ( f ( k ) ) for every k ∈ N and for every x ∈ { 0 , 1 } N (the shift map on { 0 , 1 } N is obtained by the function k ↦ k + 1 ). ( 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 , σ f ) is homeomorphic to β ( N ) . Two functions g , h : N → N are defined so that E ( { 0 , 1 } N , σ g ) is homeomorphic to the Cantor set, and E ( { 0 , 1 } N , σ h ) is the one-point compactification of N with the discrete topology.