Entropy structures with continuous partitions of unity

TL;DR

This paper introduces new definitions for various entropy and pressure concepts using continuous partitions of unity, proving a tail variational principle and extending existing notions to address open questions.

Contribution

It provides equivalent definitions for entropies and pressures via continuous partitions of unity and extends entropy structures to almost-increasing sequences.

Findings

Established a tail variational principle for the new definitions.

Extended Downarowicz's entropy structures to account for almost-increasing sequences.

Deduced a partial answer to a question posed by Newhouse.

Abstract

Using only continuous partitions of unity, we provide equivalent definitions for the metric, topological and topological tail entropies and pressures of a continuous self-map of a compact set, as well as their conditional versions. A tail variational principle for these new definitions is proved. We extend Downarowicz's notions of candidates and entropy structures to account for almost-increasing sequences of functions arising from the new definitions. Finally, we deduce a partial answer to a question raised by Newhouse.