Rank 2 proximal Cantor systems are residually scrambled

Downarowicz and Maass [7 T. Downarowicz and A. Maass, Finite-rank Bratteli–Vershik diagrams are expansive, Ergod. Th. Dynam. Sys. 28 (2008), pp. 739–747.[Web of Science ®] , [Google Scholar]] 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 E. Akin and S. Kolyada, Li–Yorke sensitivity, Nonlinearity 16(4) (2003), pp. 1421–1433.[Crossref], [Web of Science ®] , [Google Scholar]], 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.