  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.
