Iterates of dynamical systems on compact metrizable countable spaces
S. Garc\'ia-Ferreira, Y. Rodriguez-L\'opez, C. Uzc\'ategui

TL;DR
This paper investigates the structure of Ellis semigroups in dynamical systems on compact metrizable countable spaces, revealing conditions under which functions are continuous or discontinuous, and providing a specific example illustrating mixed behavior.
Contribution
It characterizes the continuity properties of Ellis semigroup functions in countable compact metric spaces and constructs an example with mixed continuity.
Findings
If all accumulation points are periodic, then all functions in $E(X,f)^*$ are either continuous or discontinuous.
An example exists where $E(X,f)^*$ contains both continuous and discontinuous functions.
The structure of $E(X,f)$ depends on the periodicity of accumulation points.
Abstract
Given a dynamical system , we let denote its Ellis semigroup and . We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a phase space. We show that if is a dynamical system such that is a compact metric countable space and every accumulation point is periodic, then either each function of is continuous or each function of is discontinuous. We describe an example of a dynamical system where is a compact metric countable space, the orbit of each accumulation point is finite and contains continuous and discontinuous functions.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals · Functional Equations Stability Results
