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Volume-Preserving Diffeomorphisms with the \(\mathcal {M}_0\)-Shadowing Properties

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Abstract

This paper studies the \(\mathcal {M}_0\)-shadowing property for two types of volume-preserving diffeomorphisms defined on compact manifolds. For symplectic diffeomorphisms defined on symplectic manifolds, the \(C^1\)-interior of the set of all symplectic diffeomorphisms with the \(\mathcal {M}_0\)-shadowing property is described by the set of the Anosov diffeomorphisms. If a volume-preserving diffeomorphism in \(\mathrm{Diff}^1_{\mu }(M)\) is a \(C^1\)-stable \(\mathcal {M}_0\)-shadowing diffeomorphism, then M admits a volume-hyperbolic dominated splitting.

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Authors and Affiliations

  1. School of Sciences, Southwest Petroleum University, Chengdu, 610500, Sichuan, People’s Republic of China

    Xinxing Wu

  2. Department of Mathematics, Shandong University, Weihai, 264209, Shandong, People’s Republic of China

    Xu Zhang

  3. School of Statistics, Qufu Normal University, Qufu, 273165, Shandong, People’s Republic of China

    Fu Sun

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  1. Xinxing Wu
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  2. Xu Zhang
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  3. Fu Sun
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Correspondence to Xu Zhang or Fu Sun.

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This work was completed with the support of the National Natural Science Foundation of China (Nos. 11601449 and 11701328), the National Nature Science Foundation of China (Key Program) (No. 51534006), Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013), Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No. 2017CXTD02), Young Scholars Program of Shandong University,Weihai (No. 2017WHWLJH09), and the Fundamental Research Funds for the Central Universities (No. 2019ZRJC005).

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Wu, X., Zhang, X. & Sun, F. Volume-Preserving Diffeomorphisms with the \(\mathcal {M}_0\)-Shadowing Properties. Mediterr. J. Math. 18, 45 (2021). https://doi.org/10.1007/s00009-020-01691-4

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  • DOI: https://doi.org/10.1007/s00009-020-01691-4

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