Relative topological entropy and relative mean dimension of induced factors

TL;DR

This paper explores the relationship between relative topological entropy and mean dimension in amenable group actions, establishing equivalences and conditions linking these invariants for factor maps and their induced factors.

Contribution

It provides new theoretical results connecting relative topological entropy and mean dimension for amenable group actions, clarifying their interplay.

Findings

Zero relative topological entropy in a factor map implies the same for the induced map.

Positive relative topological entropy corresponds to infinite relative mean dimension in the induced map.

Established equivalences deepen understanding of dynamical invariants in group actions.

Abstract

We study the relation of relative topological entropy and relative mean dimension between a factor map and its induced factor map for amenable group actions. On the one hand, we prove that a factor map has zero relative topological entropy if and only if so does the induced factor map. On the other hand, we prove that a factor map has positive relative topological entropy if and only if the induced factor map has infinite relative mean dimension.