L-shadowing for the induced hyperspace homeomorphism

TL;DR

This paper establishes an equivalence between a homeomorphism's L-shadowing property and that of its induced hyperspace homeomorphism, revealing new insights into the structure of such systems.

Contribution

It proves the if-and-only-if condition for the L-shadowing property between a homeomorphism and its induced hyperspace, and uncovers uniform convergence properties in the system.

Findings

Equivalence of L-shadowing property between homeomorphism and induced hyperspace

Existence of points with uniform convergence to zero in asymptotic structures

Contrast with non-uniform contraction in local stable/unstable sets

Abstract

We prove that a homeomorphism f of a compact metric space X satisfies the L-shadowing property if and only if its induced hyperspace homeomorphism also satisfies the L-shadowing property. In the proof, assuming only the L-shadowing property, we obtain the existence of points in the asymptotic local-product-structure with iterates approaching in a uniform rate of convergence to zero. This contrasts with the lack of uniformity of contraction on local stable/unstable sets on many homeomorphisms with the L-shadowing property.