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Topological stability for flows from a Gromov-Hausdorff viewpoint. (English) Zbl 1498.37021

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.

MSC:

37B25 Stability of topological dynamical systems
37B65 Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems
37B02 Dynamics in general topological spaces
54E40 Special maps on metric spaces
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