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arXiv · 1902.07578

Beyond topological hyperbolicity: the L-shadowing property

A. Artigue·B. Carvalho·W. Cordeiro·J. Vieitez
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Abstract

In this paper we further explore the L-shadowing property defined in [17] 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.

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A. Artigue, B. Carvalho, W. Cordeiro, J. Vieitez. 2019-02-20. Beyond topological hyperbolicity: the L-shadowing property. https://doi.org/10.1016/j.jde.2019.09.052

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