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.

Figures (3)

• Figure 1: A sequence $\tilde{w}$ corresponds to an infinite cycle of this diagram. Each $w_k=x$ or $y$, indicates that at the $k$-turn the sequence follows the orbit segment of $x$ or $y$.
• Figure 2: Illustration for the proof of Lemma \ref{['lemLshByAsint']}.
• Figure 3: A graph presenting the shift space $X_{(3,4)}$.

Theorems & Definitions (41)

• Definition
• Theorem A
• Definition
• Theorem B
• Corollary C
• Theorem D
• Theorem E
• Theorem F
• Theorem G
• Definition 2.1