Topological stability and pseudo-orbit tracing property for expansive measures (Q501621)

It is known that every expansive homeomorphism \(f\) with the pseudo-orbit tracing property of a compact metric space \((X,d)\) is topologically stable (see Theorem 4 in [\textit{P. Walters}, Lect. Notes Math. 668, 231--244 (1978; Zbl 0403.58019)]). In the present nice paper, the authors are able to give a measurable version of this result. Indeed, given a homeomorphism \(f\) from a compact metric space \(X\), Theorem 3.1 establishes that every expansive Borel measure \(\mu\) (with respect to \(f\)) having the pseudo-orbit tracing property (POTP) is topologically stable. The above main result motivates the study of other properties related with topological stability and POTP for Borel measures. Amongst others, it is proved that every nonatomic Borel measure is topologically stable with respect to any topologically stable homeomorphism; every topologically stable measure of an expansive homeomorphism is nonatomic (hence expansive); a minimal homeomorphism approximated by periodic homeomorphisms of a compact metric space has no topologically stable measures. Finally, in the last section the authors apply the previous results to the study of topologically stable measures for homeomorphisms of the circle. In this sense, it is interesting to stress the relationship with the topological dynamics of the homeomorphism. For instance, let us mention that the Dirac measure supported on a point \(p\) is topologically stable if and only if \(p\) is wandering; the Lebesgue measure of the circle is topologically stable only if there are periodic points; and there are topologically stable measures only if there are wandering points.