On some aspects of local thermodynamical formalism

TL;DR

This paper introduces a new concept called translocal entropy, which accounts for initial transients in dynamical systems, and explores its properties and relationships with existing entropy measures.

Contribution

It proposes and analyzes the translocal entropy, adjusting local topological entropy to include transient dynamics, and extends related measure-theoretic concepts.

Findings

Translocal entropy is constant for topologically transitive systems under mild conditions.

Translocal entropy can be expressed in terms of Lyapunov exponents.

The adjustment impacts measure-theoretic local entropy and pressure functions.

Abstract

In 2007, Ye \& Zhang introduced a version of local topological entropy. Since their entropy function is, as we show under mild conditions, constant for topologically transitive dynamical systems, we propose to adjust the notion in a way that does not neglect the initial transient part of an orbit. We investigate the properties of this ``transient'' version, which we call translocal entropy, and compute it in terms of Lyapunov exponents for various dynamical systems. We also investigate how this adjustment affects measure-theoretic local (Brin-Katok) entropy and local pressure functions, generalizing some partial variation principles of Ma \& Wen.