N-expansive homeomorphisms with the shadowing property

Abstract

We discuss the dynamics of $n$-expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every $n\in \mathbb{N}$, we exhibit an $n$-expansive homeomorphism, which is not $(n-1)$-expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of $n$-expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing $n$-expansive homeomorphisms that are not expansive.

References in corpus (1)

• Product Anosov diffeomorphisms and the two-sided limit shadowing property

Cited by in corpus (12)

• Beyond topological hyperbolicity: the L-shadowing property
• On specification and measure expansiveness
• Positively n-expansive homeomorphisms and the L-shadowing property
• Continuum-wise hyperbolicity
• Measure Expansivity and Specification for Pointwise Dynamics
• Countably and entropy expansive homeomorphisms with the shadowing property
• Positive Entropy Through Pointwise Dynamics
• N-expansive Flows
• Suspensions of homeomorphisms with the two-sided limit shadowing property
• On The Entropy of Continuous Flows With Uniformly Expansive Points and The Globalness of Shadowable Points With Gaps
• Product Anosov diffeomorphisms and the two-sided limit shadowing property
• Stability Theorems for Group Actions on Uniform Spaces