The Ellis semigroup of a nonautonomous discrete dynamical system

TL;DR

This paper extends the classical Ellis semigroup concept to nonautonomous discrete dynamical systems on compact metric spaces, providing a detailed description and exploring its properties and connections to system topology.

Contribution

It introduces the Ellis semigroup for nonautonomous systems, extending classical theory and offering a new framework for analyzing their topological dynamics.

Findings

Provides a precise description of the Ellis semigroup using ultrafilter convergence.

Shows the connection between the semigroup's properties and the system's topological features.

Extends classical Ellis semigroup theory to nonautonomous systems.

Abstract

We introduce the {\it Ellis semigroup} of a nonautonomous discrete dynamical system $(X,f_{1,\infty})$ when $X$ is a metric compact space. The underlying set of this semigroup is the pointwise closure of $\{f\sp{n}_1 \, |\, n\in \mathbb{N}\}$ in the space $X\sp{X}$. By using the convergence of a sequence of points with respect to an ultrafilter it is possible to give a precise description of the semigroup and its operation. This notion extends the classical Ellis semigroup of a discrete dynamical system. We show several properties that connect this semigroup and the topological properties of the nonautonomous discrete dynamical system.