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- W1989281570 abstract "Given a dynamical system (X,f) with X a compact metric space and a free ultrafilter p on N, we define fp(x)=p-limn→∞fn(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 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:{0,1}N→{0,1}N by σf(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). 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,σf) is homeomorphic to β(N). Two functions g,h:N→N are defined so that E({0,1}N,σg) is homeomorphic to the Cantor set, and E({0,1}N,σh) is the one-point compactification of N with the discrete topology." @default.
- W1989281570 created "2016-06-24" @default.
- W1989281570 creator A5063956965 @default.
- W1989281570 date "2012-04-01" @default.
- W1989281570 modified "2023-09-30" @default.
- W1989281570 title "Dynamical properties of certain continuous self maps of the Cantor set" @default.
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- W1989281570 doi "https://doi.org/10.1016/j.topol.2011.06.062" @default.
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