Non-uniform Cocycles for Some Uniquely Ergodic Minimal Dynamical Systems on Connected Spaces

TL;DR

This paper investigates the existence of non-uniform cocycles in certain uniquely ergodic minimal dynamical systems on connected spaces, classifying systems and providing affirmative answers for specific classes.

Contribution

It classifies uniquely ergodic minimal systems into three types and proves the existence of non-uniform cocycles for the first two classes, including systems with arbitrary topological entropy.

Findings

Affirmative answer for non-uniform cocycles in not totally uniquely ergodic systems.

Existence of such systems with arbitrary topological entropy.

Classification of systems into three distinct ergodic and mixing types.

Abstract

In this paper, we pay attention to a weaker version of Walters's question on the existence of non-uniform cocycles for uniquely ergodic minimal dynamical systems on non-degenerate connected spaces. We will classify such dynamical systems into three classes: not totally uniquely ergodic; totally uniquely ergodic but not topological weakly mixing; totally uniquely ergodic and topological weakly mixing. We will give an affirmative answer to such question for the first two classes. Also, we will show the existence of such dynamical systems in the first class with arbitrary topological entropy.