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 Hölder 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 $2\times 2$ aperiodic transition matrix. Moreover, we show that a ‘non-rigidity’ result holds for a certain topological Markov shift with a $3\times 3$ aperiodic transition matrix.

摘要

在本文中，我们考虑拓扑马尔可夫位移的熵谱。我们证明，如果两个具有 Hölder 连续势的吉布斯测度的保测动力系统是同构的，那么它们的熵谱是相同的。这个结果提出了一个新的刚性问题。我们称这个问题为弱刚性问题，与 Barreira 和 Saraiva 提出的强刚性问题形成对比。我们给出了拓扑马尔可夫位移上马尔可夫测度的弱刚性问题的完整答案$2\times 2$非周期转移矩阵。此外，我们证明了“非刚性”结果适用于某个具有$3\times 3$ 非周期转移矩阵。