On the genericity of the shadowing property for conservative homeomorphisms

TL;DR

This paper proves that shadowing and related properties are generically present in conservative and dissipative homeomorphisms on compact manifolds, highlighting their typical behavior in these dynamical systems.

Contribution

It establishes the genericity of shadowing, periodic shadowing, and several related properties for both conservative and dissipative homeomorphisms on topological manifolds.

Findings

Shadowing property is generic for conservative homeomorphisms.

Periodic shadowing property is also generic in this setting.

The specification and average shadowing properties are generically valid for conservative systems.

Abstract

We prove the genericity of the shadowing and periodic shadowing properties for both conservative and dissipative homeomorphisms on a compact connected manifold. Our proof is valid for topological manifolds and still holds in the dissipative case. As a consequence of this result, we establish the genericity of the specification property, the average shadowing property and the asymptotic average shadowing property in the conservative case.