Non-homeomorphic topological rank and expansiveness

Downarowicz and Maass (2008) have shown that every Cantor minimal homeomorphism with finite topological rank $K > 1$ is expansive. Bezuglyi, Kwiatkowski and Medynets (2009) extended the result to non-minimal cases. On the other hand, Gambaudo and Martens (2006) had expressed all Cantor minimal continuou surjections as the inverse limit of graph coverings. In this paper, we define a topological rank for every Cantor minimal continuous surjection, and show that every Cantor minimal continuous surjection of finite topological rank has the natural extension that is expansive.