Skew product dynamical systems, Ellis groups and topological centre

TL;DR

This paper introduces a new class of skew product dynamical systems called Milnes type systems, explores their ergodic and topological properties, and computes the topological centre of their Ellis groups, extending previous work by Hahn.

Contribution

It provides a general construction of skew product systems, including Hahn's as a special case, and analyzes their properties and Ellis group structures.

Findings

Milnes type systems are natural extensions of systems related to distal functions.

The topological centre of the Ellis group for these systems is explicitly calculated.

The construction generalizes previous skew product systems studied by Hahn.

Abstract

In this paper, a general construction of a skew product dynamical system, for which the skew product dynamical system studied by Hahn is a special case, is given. Then the ergodic and topological properties (of a special type) of our newly defined systems (called Milnes type systems) are investigated. It is shown that the Milnes type systems are actually natural extensions of dynamical systems corresponding to some special distal functions. Finally, the topological centre of the Ellis group of any skew product dynamical system is calculated.