Continuity of isomorphisms applied to rigidity problems of entropy spectra

For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all 2-locally constant functions $f$ on the shift such that all those isomorphisms defined on the system associated with $f$ are induced from automorphisms of the shift. We prove that this set contains a full-measure open set of the space of all 2-locally constant functions on the shift. We apply this result to rigidity problems of entropy spectra and show that the strong non-rigidity occurs if and only if so does the weak non-rigidity.