Beyond topological hyperbolicity: The L-shadowing property

In this paper we further explore the L-shadowing property defined in [20] for dynamical systems on compact spaces. We prove that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere satisfy this property. Homeomorphisms satisfying the L-shadowing p...

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Bibliographic Details
Published in:Journal of Differential Equations 2020-03, Vol.268 (6), p.3057-3080
Main Authors: Artigue, Alfonso, Carvalho, Bernardo, Cordeiro, Welington, Vieitez, José
Format: Article
Language:English
Subjects:
Expansiveness
Hyperbolicity
L-shadowing
Topological
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Summary:In this paper we further explore the L-shadowing property defined in [20] for dynamical systems on compact spaces. We prove that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere satisfy this property. Homeomorphisms satisfying the L-shadowing property have a spectral decomposition where the basic sets are either expansive or contain arbitrarily small topological semi-horseshoes (periodic sets where the restriction is semiconjugate to a shift). To this end, we characterize the L-shadowing property using local stable and unstable sets and the classical shadowing property. We exhibit homeomorphisms with the L-shadowing property and arbitrarily small topological semi-horseshoes without periodic points. At the end, we show that positive finite-expansivity jointly with the shadowing property imply that the space is finite.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2019.09.052