Weak rigidity of entropy spectra

摘要

In this paper, we consider entropy spectra on topological Markov shifts. We prove that if two measure-preserving dynamical systems of Gibbs measures with Hoelder 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 a 2 × 2 aperiodic transition matrix. Moreover, we show that a 'non-rigidity' result holds for a certain topological Markov shift with a 3 × 3 aperiodic transition matrix.