Topological rank does not increase by natural extension of Cantor minimals

TL;DR

This paper demonstrates that the topological rank of Cantor minimal continuous surjections remains unchanged under natural extensions, with implications for minimal symbolic systems.

Contribution

It proves that natural extensions of Cantor minimal continuous surjections do not increase their topological ranks, extending previous notions to a broader class of systems.

Findings

Topological rank remains invariant under natural extensions.

Application to minimal symbolic systems confirms the invariance.

Supports the robustness of topological rank in dynamical systems.

Abstract

Downarowicz and Maass (2008) have defined the topological rank for all Cantor minimal homeomorphisms. On the other hand, Gambaudo and Martens (2006) 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 the minimal symbolic case.