Proximal Cantor systems with topological rank 2 are residually scrambled

TL;DR

This paper studies zero-dimensional proximal Cantor systems with topological rank 2, showing they are residually scrambled and analyzing their ergodic measures, including conditions for unique ergodicity and examples with various mixing properties.

Contribution

It establishes that all proximal Cantor systems with topological rank 2 are residually scrambled and characterizes their ergodic measures, including criteria for unique ergodicity.

Findings

All proximal Cantor systems with topological rank 2 are residually scrambled.

Such systems have at most two ergodic measures.

Conditions for unique ergodicity are provided.

Abstract

Downarowicz and Maass (2008) proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of all essentially minimal zero-dimensional systems. We consider the cases in which topological ranks are 2 and unique minimal sets are fixed points. Akin and Kolyada (2003), in their study of Li--Yorke sensitivity, showed that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. However, a finite topological rank implies expansiveness; furthermore, in the case of proximal Cantor systems with topological rank 2, the expansiveness is always from the lowest degree. Rank 2 zero-dimensional systems might be thought as a part of the rank 1 transformations that are considered in the vast field of ergodic theory. However, these systems are also interesting from the perspective of topological chaos theory; e.g., in this study, we show that all proximal Cantor systems with topological rank 2 are residually scrambled. In addition, we investigate the finite invariant measures on these systems. Evidently, such systems have at most two ergodic measures. We present a necessary and sufficient condition for the unique ergodicity of these systems. In addition, we show that the number of ergodic measures of systems that are topologically mixing can be 1 and 2. Moreover, we present examples that are topologically weakly mixing, not topologically mixing, and uniquely ergodic. Finally, we show that the number of ergodic measures of systems that are not weakly mixing can be 1 and 2.