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Diffeomorphisms with a generalized Lipschitz shadowing property

  • * Corresponding author: Xiao Wen

    * Corresponding author: Xiao Wen
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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  • 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.

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


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  • [1] S-T. Liao, Obstruction sets I, Acta Math. Sinica., 23 (1980), 411-453. 
    [2] R. Mañé, Characterizations of AS diffeomorphisms, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Springer, Berlin, 597 (1977), 389–394.
    [3] R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161-210. 
    [4] S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706. Springer-Verlag, Berlin, 1999.
    [5] S. Y. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515.  doi: 10.1088/0951-7715/23/10/009.
    [6] S. Y. Pilyugin and K. Sakai, Shadowing and Hyperbolicity, Lecture Notes in Math., 2193. Springer, Cham, 2017. doi: 10.1007/978-3-319-65184-2.
    [7] V. A. Pliss, Bounded solutions of nonhomogeneous linear systems of differential equations, Problems in the Asymptotic Theory of Nonlinear Oscillations [in Russian], Naukova Dumka, Kiev, 278 (1977), 168–173.
    [8] C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425-437.  doi: 10.1216/RMJ-1977-7-3-425.
    [9] K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386. 
    [10] D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187-4205.  doi: 10.3934/dcds.2013.33.4187.
    [11] L. Wen, Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity, Graduate Studies in Mathematics 173, American Mathematical Society, Providence, RI, 2016. doi: 10.1090/gsm/173.
    [12] X. WenS. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations., 246 (2009), 340-357.  doi: 10.1016/j.jde.2008.03.032.
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