Topological Rank Does Not Increase by Natural Extension of Cantor Minimals

Abstract

Abstract. Downarowicz and Maass have defined the topological rank for all Cantor minimal homeomorphisms. On the other hand, Gambaudo and Martens have expressed all Cantor minimal continuous surjections as the inverse limits of certain graph coverings. Using the aforementioned results, we previously extended the notion of topological rank to all Cantor minimal continuous surjections. In this paper, we show that taking natural extensions of Cantor minimal continuous surjections does not increase their topological ranks. Further, we apply the result to some minimal symbolic cases.

All minimal Cantor systems are slow

“…In this section, we recall notions from the theory of graph covers that we are going to use in the proof of Theorem . For more on graph covers, and various recent applications see, for example, .…”

“…A description of all (not necessarily invertible) minimal zero‐dimensional dynamical systems in terms of inverse limits of graph covers of particular type was given by Gambaudo and Martens . Following Shimomura , we refer to them as coverings of Gambaudo–Martens type (GM‐coverings for short). Let be a sequence of graph covers , where is a graph homomorphism between graphs and .…”