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.
Highlights
The geometric Lorenz attractor is one of the most important examples for the flow theory
In this article we show that this no-shadowing phenomenon appears for a wide class of “nearly-hyperbolic” sets, namely, the so called singular hyperbolic sets, or more generally, the so called multisingular hyperbolic sets
We will show that every chain transitive multisingular hyperbolic set with a singularity does not admit the shadowing property
Summary
The geometric Lorenz attractor is one of the most important examples for the flow theory. It shares many important properties with a hyperbolic set while itself is not a hyperbolic set It is robustly transitive, robustly with periodic orbits dense, etc., unlike a hyperbolic set, it does not admit the robust shadowing property. In this article we show that this no-shadowing phenomenon appears for a wide class of “nearly-hyperbolic” sets, namely, the so called singular hyperbolic sets (among which the geometric Lorenz attractor is a particular example), or more generally, the so called multisingular hyperbolic sets. We will show that every chain transitive multisingular hyperbolic set with a singularity does not admit the shadowing property. This highlights the striking and delicate difference between flows with a singularity and flows without singularities.
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