Measure theoretical entropy of covers

TL;DR

This paper introduces and proves the equivalence of three measure-theoretic entropy notions for covers in dynamical systems, clarifying their relationships and recovering key variational principles.

Contribution

It defines a new entropy notion for ergodic systems and proves the equivalence of three existing notions, answering an open question and linking covers to partitions.

Findings

The three entropy notions coincide in measure-theoretic dynamical systems.

The new entropy notion is defined specifically for ergodic systems.

The results recover classical variational principles and inequalities.

Abstract

In this paper we introduce three notions of measure theoretical entropy of a measurable cover U in a measure theoretical dynamical system. Two of them were already introduced in [R] and the new one is defined only in the ergodic case. We then prove that these three notions coincide, thus answering a question posed in [R] and recover a variational inequality (proved in [GW]) and a proof of the classical variational principle based on a comparison between the entropies of covers and partitions.