Continuity of isomorphisms applied to rigidity problems of entropy spectra

TL;DR

This paper investigates the continuity of isomorphisms in measure-preserving dynamical systems derived from topological Markov shifts, revealing conditions under which entropy spectra exhibit rigidity or non-rigidity.

Contribution

It establishes that a full-measure open set of 2-locally constant functions leads to isomorphisms induced by automorphisms, advancing understanding of entropy spectrum rigidity.

Findings

Full-measure open set of functions with automorphism-induced isomorphisms

Rigidity of entropy spectra linked to non-rigidity conditions

Strong and weak non-rigidity are equivalent phenomena

Abstract

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.