Weak rigidity of entropy spectra

TL;DR

This paper investigates the weak rigidity of entropy spectra in topological Markov shifts, proving that isomorphic Gibbs measure systems have identical spectra and providing a complete solution for a specific case.

Contribution

It introduces the weak rigidity problem for entropy spectra and offers a complete solution for Markov measures with a 2x2 aperiodic transition matrix.

Findings

Isomorphic Gibbs measure systems have identical entropy spectra.

Complete solution provided for 2x2 aperiodic transition matrices.

Raises new questions about rigidity in dynamical systems.

Abstract

In this paper, we consider entropy spectra on topological Markov shifts. We prove that if two measure-preserving dynamical systems of Gibbs measures with H\"older continuous potentials are isomorphic, then their entropy spectra are the same. This result raises a new rigidity problem. We call this problem the weak rigidity problem, contrasting it with the strong rigidity problem proposed by Barreira and Saraiva. We give a complete answer to the weak rigidity problem for Markov measures on a topological Markov shift with an aperiodic transition matrix of size 2.