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 Lshadowing 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.

Suspensions of homeomorphisms with the two-sided limit shadowing property

“…It is called the two-sided limit shadowing property. The dynamics of systems with such property has been studied (see [5], [6], [7] and [10]) and the class of homeomorphisms satisfying it is growing (see [2], [8] and [9]). It is known that the two-sided limit shadowing property differs in several ways from the limit shadowing property.…”

“…For example, systems with the two-sided limit shadowing property must have the shadowing property, the average shadowing property, the asymptotic average shadowing property, must be topologically mixing, admit the specification property and have positive topological entropy (see [10] for the proofs) making it one of the strongest known notions of pseudo-orbit tracing properties, while there are systems with the limit shadowing property but without any of these dynamical properties (see [12] for example). Examples of homeomorphisms satisfying the twosided limit shadowing property: topologically mixing Anosov diffeomorphisms (and more generally topologically hyperbolic homeomorphisms [6]), pseudo-Anosov diffeomorphisms of the two-dimensional sphere [2] and some wild examples of infinite products of subshifts [10] without periodic points.…”

Flows with the weak two-sided limit shadowing property

“…The theory of shadowing in dynamical systems is a rapidly developing branch of modern global theory of dynamical systems. There are various notions of shadowing for dynamical systems (homeomorphisms or flows) on compact metric spaces, and a lot of interesting results of dynamical systems with various type of shadowing were investigated in many papers (e.g., see [2,4,6,7,9,11,13]).…”

“…Very recently, Artigue et al [4] introduced the notion of L-shadowing for homeomorphisms on compact metric spaces, and explored the dynamics of homeomorphisms with the L-shadowing property.…”

“…Important dynamics of the homeomorphisms with the L-shadowing property were investigated by Artigue et al in [4] recently.…”