Diffeomorphisms with various $C^1$ stable properties

TL;DR

This paper characterizes when robustly transitive invariant sets on smooth compact manifolds exhibit stable shadowing, transitive specification, or barycenter properties, showing these are equivalent to the sets being hyperbolic basic sets.

Contribution

It establishes a precise equivalence between $C^1$-stability of certain dynamical properties and hyperbolic basic sets for robustly transitive sets, extending to volume-preserving systems.

Findings

Robustly transitive sets with stable shadowing are hyperbolic basic sets.

Stable transitive specification property characterizes hyperbolic basic sets.

Results apply to both volume-preserving and general cases.

Abstract

Let $M$ be a smooth compact manifold and $\Lambda$ be a compact invariant set. In this paper we prove that for every robustly transitive set $\Lambda$, $f|_\Lambda$ satisfies a $C^1-$generic-stable shadowable property (resp., $C^1-$generic-stable transitive specification property or $C^1-$generic-stable barycenter property) if and only if $\Lambda$ is a hyperbolic basic set. In particular, $f|_\Lambda$ satisfies a $C^1-$stable shadowable property (resp., $C^1-$stable transitive specification property or $C^1-$stable barycenter property) if and only if $\Lambda$ is a hyperbolic basic set. Similar results are valid for volume-preserving case.