We prove that every finite-expansive homeomorphism with the shadowing property has a kind of stability. This stability will be good enough to imply both the shadowing property and the denseness of periodic points in the chain recurrent set. Next we analyze the N-shadowing property which is really stronger than the multishadowing property in Cherkashin and Kryzhevich (Topol Methods Nonlinear Anal 50(1): 125–150, 2017). We show that an equicontinuous homeomorphism has the N-shadowing property for some positive integer N if and only if it has the shadowing property. © 2021, Sociedade Brasileira de Matemática.

We discuss the stability and the expansivity of the tent map f: [0, 1] ? [0, 1] defined by f(x) = 2 min{x, 1 ? x} for 0 ? x ? 1. Indeed, we show that f is neither topologically stable nor orbit shift topologically stable nor countably-expansive but is cw-topologically stable, orbit shift cw-expansive, and orbit shift ?-persistent. © 2020 American Mathematical Society

The authors would like to thank Carlos A. Morales, for useful talks on the subject of expansiveness and shadowing. Also, the authors would like to thank Professor H. Miranda and the anonymous reviewers for their valuable comments that helped to improve the final version of the article. The first author was partially supported by CONICYT PFCHA/DOCTORADO NACIONAL/2017-21170110 and Agencia Nacional de Investigaci?n y Desarrollo-ANID, Chile, project FONDECYT 1181061. The second author was partially supported by Agencia Nacional de Investigaci?n y Desarrollo-ANID, Chile, project FONDECYT 1181061, by Universidad del B?o-B?o, Chile, project 196108 GI/C, and by Programa do Postdoutorado Ver?o 2017-2019, IMPA, Rio de Janeiro, Brasil. The last author was partially supported by FONDECYT (Per?) contract 100?2018.