# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020346

## 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

Received  January 2020 Revised  August 2020 Published  October 2020

Fund Project: 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

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.

Citation: Manseob Lee, Jumi Oh, Xiao Wen. Diffeomorphisms with a generalized Lipschitz shadowing property. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020346
##### References:
 [1] S-T. Liao, Obstruction sets I, Acta Math. Sinica., 23 (1980), 411-453.   Google Scholar [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.  Google Scholar [3] R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161-210.   Google Scholar [4] S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706. Springer-Verlag, Berlin, 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.   Google Scholar [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.  Google Scholar [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.  Google Scholar [12] X. Wen, S. 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.  Google Scholar

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##### References:
 [1] S-T. Liao, Obstruction sets I, Acta Math. Sinica., 23 (1980), 411-453.   Google Scholar [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.  Google Scholar [3] R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161-210.   Google Scholar [4] S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706. Springer-Verlag, Berlin, 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.   Google Scholar [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.  Google Scholar [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.  Google Scholar [12] X. Wen, S. 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.  Google Scholar
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