Entropy spectrum of rotation classes

TL;DR

This paper investigates the entropy spectrum of rotation classes for collections of continuous potentials in dynamical systems, demonstrating that these spectra are often maximal and providing criteria for this maximality.

Contribution

It establishes the maximality of entropy spectra for broad classes of systems and potentials, offering new insights into the structure of invariant measures in dynamics.

Findings

Entropy spectra are maximal for large classes of systems and potentials.

Criteria are provided for the maximality of ergodic entropy spectra.

Results complement the classical Riesz representation theorem in dynamics.

Abstract

In this note we study the entropy spectrum of rotation classes for collections of finitely many continuous potentials $\varphi_1,\dots,\varphi_m:X\to \mathbb{R}$ with respect to the set of invariant measures of an underlying dynamical system $f:X\to X$. We show for large classes of dynamical systems and potentials that these entropy spectra are maximal in the sense that every value between zero and the maximum is attained. We also provide criteria that imply the maximality of the ergodic entropy spectra. For $m$ being large, our results can be interpreted as a complimentary result to the classical Riesz representation theorem in the dynamical context.