Rank 2 proximal Cantor systems are residually scrambled

ABSTRACT

Downarowicz and Maass [7] 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 the 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 [2], had shown that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. The 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 proximal Cantor systems are residually scrambled. 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 the 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 the systems that are not weakly mixing can be 1 and 2.

Keywords:

• Proximal
• topological rank
• ergodic
• symbolic dynamics
• residually scrambled

MSC 2010 CLASSIFICATION:

• Primary 37B05
• 37B10
• 54H20

Acknowledgments

The author would like to thank the anonymous referee(s) for their pertinent advice, because of which this paper has improved. The author would like to thank Editage (www.editage.jp) for providing English-language editing services. This work was partially supported by JSPS KAKENHI (Grant Number 16K05185).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Japan Society for the Promotion of Science [grant number 16K05185].