Stably weakly shadowing transitive sets and dominated splittings

TL;DR

This paper proves that in dynamical systems, any transitive set with stable weak shadowing property is either a sink, a source, or has a dominated splitting, clarifying the structure of such sets.

Contribution

It establishes a dichotomy for $C^1$-stably weakly shadowing transitive sets, showing they are either sinks, sources, or admit a dominated splitting, which is a new structural insight.

Findings

Transitive sets with stable weak shadowing are either sinks or sources.

Such sets admit a dominated splitting.

Provides a classification of these sets in dynamical systems.

Abstract

We prove that for any $C^1$-stably weakly shadowing transitive set $\Lambda$, either $\Lambda$ is a sink or a source, or $\Lambda$ admits a dominated splitting.