Topological stability for flows from a Gromov-Hausdorff viewpoint (Q2115188)

The author introduces a notion of Gromov-Hausdorff distance between two flows of possibly different metric spaces. The notion is applied to define a type of topological stability for a flow \(\phi\) on a compact metric space \(X\) as follows. A flow \(\phi\) is said to be \(\sigma\)-topologically GH-stable if for every \(\varepsilon >0\) there is \(\delta >0\) such that for any flow \(\psi\) on a compact metric space \(Y\) with \(D_{GH^0}(\phi, \psi) < \delta\), there is an \(\varepsilon\)-isometry \(h: Y \rightarrow X\) which is continuous on a residual subset \(Y\), and takes orbits of \(\psi\) to orbits of \(\phi\). The main result in this paper states that any expansive flow of a compact metric space with the pseudo-orbit tracing property is \(\sigma\)-topologically GH-stable.