Beyond topological hyperbolicity: the L-shadowing property

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.

21 pages, 3 figures

Cited by in corpus (10)

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• Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms
• Continuum-wise hyperbolic homeomorphisms on surfaces
• Suspensions of homeomorphisms with the two-sided limit shadowing property
• Sensitivity, local stable/unstable sets and shadowing
• First-time sensitive homeomorphisms
• Shadowing maps
• Stable/unstable continua of cw-expansive flows
• Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy