Diffeomorphisms with a generalized Lipschitz shadowing property

Manseob Lee; Jumi Oh; Xiao Wen

doi:10.3934/dcds.2020346

Article Contents

2021, Volume 41, Issue 4: 1913-1927. Doi: 10.3934/dcds.2020346

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Diffeomorphisms with a generalized Lipschitz shadowing property

1.
Department of Mathematics, Mokwon University, Daejeon 35349, Korea
2.
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
3.
School of Mathematical Sciences, Beihang University, Beijing 100191, China

^* Corresponding author: Xiao Wen
^* Corresponding author: Xiao Wen

Received: December 31, 2019

Revised: July 31, 2020

Early access: October 2020

Published: April 2021

M. Lee was supported by NRF-2017R1A2B4001892 and NRF-2020R1F1A1A01051370. J. Oh was supported by NRF-2019R1A2C1002150. X. Wen was supported by National Natural Science Foundation of China (No. 11671025 and No. 11571188) and Fundamental Research Funds for the Central Universities

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Abstract

Shadowing property and structural stability are important dynamics with close relationship. Pilyugin and Tikhomirov proved that Lipschitz shadowing property implies the structural stability[5]. Todorov gave a similar result that Lipschitz two-sided limit shadowing property also implies structural stability for diffeomorpshisms[10]. In this paper, we define a generalized Lipschitz shadowing property which unifies these two kinds of Lipschitz shadowing properties, and prove that if a diffeomorphism $ f $ of a compact smooth manifold $ M $ has this generalized Lipschitz shadowing property then it is structurally stable. The only if part is also considered.

Keywords:

Perron property,
generalized Lipschitz shadowing property,
hyperbolicity,
structural stability.

Mathematics Subject Classification: Primary: 37C20, 37C05, 37D05.

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References

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