  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.
