A shadowable chain recurrent set with an attached hyperbolic singularity

TL;DR

This paper constructs a smooth flow on a 4D sphere with a hyperbolic singularity that still exhibits the shadowing property, countering a previous conjecture about such dynamical systems.

Contribution

It provides a counterexample to a conjecture by demonstrating a flow with a hyperbolic singularity that preserves the shadowing property.

Findings

Constructed a $C^ abla$-flow on a 4D sphere with a hyperbolic singularity.

Showed the flow's nonwandering set contains an attached hyperbolic singularity.

Proved the flow still has the standard shadowing property despite the singularity.

Abstract

We prove that every factor map between topological flows preserves the standard shadowing property if it is injective except for a closed orbit that shrinks to a singularity. As an application, we construct a $C^\infty$-flow on a four-dimensional sphere whose nonwandering set contains an attached hyperbolic singularity yet possesses the standard shadowing property. This gives a counterexample to a conjecture given by Arbieto, L\'{o}pez, Rego and S\'{a}nchez (Math. Annalen 390:417-437).