Relative topological conditional entropy and a Ledrappier's type variational principle for it

Entropies for factor maps of amenable group actions

“…In fact, asymptotical h-expansiveness also ensures the existence of measures with maximal entropy for amenable group actions with finite topological entropy, which follows from results obtained in [4,25] (see also the recent results proved in [9,10] in the framework of symbolic extension theory of amenable group actions). Thus it is natural to ask if we could provide such a nontrivial condition for the existence of measures with maximal relative entropy for factor maps of amenable group actions as we did for factor maps between Z-actions in [24]. In the second part of the paper we shall answer the question affirmatively.…”

“…In order to provide a nontrivial condition for the existence of invariant measures with maximal relative entropy for factor maps between amenable group actions, in this section along the line of Misiurewicz in [18] we introduce relative topological tail entropy for factor maps between amenable group actions, as we did for Z-actions in [24]. We prove a Ledrappier's type variational principle theorem 4.3 concerning relative topological tail entropy, consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which does provide such a nontrivial condition.…”

“…In order to provide such a nontrivial condition for factor maps between Z-actions, similarly to the asymptotical h-expansiveness for Z-actions [18], the same authors of this paper introduced the concept of relative topological tail entropy 3 as a generalization of topological tail entropy, and proved a Ledrappier's type variational principle relating relative topological tail entropy which generalizes the variational principle obtained by Ledrappier in [15]; consequently, there always exist measures with maximal relative entropy if a factor map has zero relative topological tail entropy. See [24] for more details.…”

“…It was used the name of relative topological conditional entropy by the authors in[24]. In order to distinguish from the concept of conditional (topological) entropy for factor maps between Z-actions introduced by Boyle et al in[3], which was extended as conditional topological entropy for factor maps of amenable group actions by Zhu in[26] (we shall recall and discuss it later), alternatively we call it here relative topological tail entropy.…”