Finite-expansivity and \(N\)-shadowing (Q2115178)

The authors introduce a kind of stability, called finite-topological stability, for a homeomorphism on a compact metric space, and show that any finite-expansive homeomorphism with the shadowing property of a compact metric space is finite-topologically stable. Moreover, if a homeomorphism on a closed manifold \(M\) with \(\dim M\geq 2\) is finite-topologically stable, then it has the shadowing property and its periodic points are dense in the chain recurrent set. Furthermore, they study the notion of \(N\)-shadowing property, which is really stronger than the multishadowing property in [\textit{D. Cherkashin} and \textit{S. Kryzhevich}, Topol. Methods Nonlinear Anal. 50, No. 1, 125--150 (2017; Zbl 1407.37008)]. They show that an equicontinuous homeomorphism has the \(N\)-shadowing property for some \(N \in \mathbb{N}\) if and only if it has the shadowing property.