Beyond topological hyperbolicity: the L-shadowing property (Q2288041)

The authors investigate the L-shadowing property for discrete dynamical systems. This concept was introduced in [\textit{B. Carvalho} and \textit{W. Cordeiro}, J. Dyn. Differ. Equations 31, No. 2, 1005--1016 (2019; Zbl 1415.37013)]. They first characterize the L-shadowing property using local stable and unstable sets and the shadowing property. In particular, they prove a spectral decomposition for homeomorphisms with L-shadowing property. More precisely, they prove that if a homeomorphism \(f\) on a compact metric space \(X\) has the L-shadowing property, then the nonwandering set can be decomposed as a disjoint union of finitely many transitive, invariant, compact sets \(B\) such that each \(B\) is either expansive or contains arbitrary small topological semi-horseshoes. Moreover, they show that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere satisfy the L-shadowing property.