No-shadowing for singular hyperbolic sets with a singularity

Xiao Wen; Lan Wen

doi:10.3934/dcds.2020258

Article Contents

2020, Volume 40, Issue 10: 6043-6059. Doi: 10.3934/dcds.2020258

This issue Previous Article Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems Next Article Integrability of moduli and regularity of denjoy counterexamples

No-shadowing for singular hyperbolic sets with a singularity

Xiao Wen^1, and
Lan Wen^2,

1.
School of Mathematical Sciences, Beihang University, Beijing 100191, China
2.
School of Mathematical Sciences, Peking University, Beijing 100871, China

Received: February 29, 2020

Published: June 2020

The first author is supported by National Natural Science Foundation of China (No. 11671025 and No. 11571188) and the Fundamental Research Funds for the Central Universities. The second author is supported by National Natural Science Foundation of China (No. 11231001)

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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.

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Mathematics Subject Classification: Primary: 37C10, 37D30, 37C50.

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