The Ellis semigroup of a nonautonomous discrete dynamical system

Abstract

We introduce the Ellis semigroup of a nonautonomous discrete dynamical system (X, f₁,∞) when X is a metric compact space. The underlying set of this semigroup is the pointwise closure of {fⁿ₁| n ∈ N) in the space X^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.

Keywords: Free ultrafilter, discrete dynamical system, nonautonomous discrete dynamical system, Ellis semigroup, compact metric space, p-limit point