Entropies for factor maps of amenable group actions

Abstract

In this paper we study various entropies for factor maps of amenable group actions. We prove firstly theorem inequalities linking relative topological entropy and conditional topological entropy (for factor maps of amenable group actions) without any additional assumption, which strengthens conditional variational principles (Zhu 2021 Nonlinearity 34 5163–85, theorems 2.12 and 3.9) proved by Zhu under additional assumptions. Then along the line of Misiurewicz (1976 Studia Math. 55 175–200), we introduce a new invariant called relative topological tail entropy and prove a Ledrappier’s type variational principle concerning it (for factor maps of amenable group actions); consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which provides a nontrivial sufficient condition for the existence of invariant measures with maximal relative entropy in the setting of factor maps of amenable group actions.