No-shadowing for singular hyperbolic sets with a singularity

TL;DR

This paper proves that singular hyperbolic chain transitive sets with singularities lack the shadowing property, and explores implications for star flows and multisingular hyperbolic sets regarding hyperbolicity and Axiom A conditions.

Contribution

It establishes the absence of shadowing in singular hyperbolic sets with singularities and links shadowing properties to hyperbolicity and Axiom A conditions in dynamical systems.

Findings

Singular hyperbolic chain transitive sets with singularities do not admit shadowing.

Star flows with shadowing on chain recurrent sets satisfy Axiom A and no-cycle conditions.

Multisingular hyperbolic sets with shadowing are hyperbolic.

Abstract

We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.