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Downarowicz and Maass (2008) proposed topological ranks for all homeomorphicCantor minimal dynamical systems using properly ordered Bratteli diagrams. Inthis study, we adopt this definition to the case of all essentially minimalzero-dimensional systems. We consider the cases in which topological ranks are2 and unique minimal sets are fixed points. Akin and Kolyada (2003), in theirstudy of Li--Yorke sensitivity, showed that if the unique minimal set of anessentially minimal system is a fixed point, then the system must be proximal.However, a finite topological rank implies expansiveness; furthermore, in thecase of proximal Cantor systems with topological rank 2, the expansiveness isalways from the lowest degree.  Rank 2 zero-dimensional systems might be thought as a part of the rank 1transformations that are considered in the vast field of ergodic theory.However, these systems are also interesting from the perspective of topologicalchaos theory; e.g., in this study, we show that all proximal Cantor systemswith topological rank 2 are residually scrambled. In addition, we investigatethe finite invariant measures on these systems. Evidently, such systems have atmost two ergodic measures. We present a necessary and sufficient condition forthe unique ergodicity of these systems. In addition, we show that the number ofergodic measures of systems that are topologically mixing can be 1 and 2.Moreover, we present examples that are topologically weakly mixing, nottopologically mixing, and uniquely ergodic. Finally, we show that the number ofergodic measures of systems that are not weakly mixing can be 1 and 2.

Proximal Cantor systems with topological rank 2 are residually scrambled