Bo Han
,
Manseob Lee
. A GENERALIZED LIPSCHITZ SHADOWING PROPERTY FOR FLOWS*[J]. Acta mathematica scientia, Series B, 2023
, 43(1)
: 259
-288
.
DOI: 10.1007/s10473-023-0115-6
[1] Abdenur F, Díaz L J.Pseudo-orbit shadowing in the C1-topology. Discrete Contin Dyn Syst, 2007, 17: 223-245
[2] Anosov D V.Geodesic flows on closed Riemannian manifolds with negative curvature. Proc Steklov Institute of Mathematics, 1967, 90
[3] Bowen R.On Axiom A Diffeomorphisms. Regional Conference Series in Mathematics, 35. Providence, RI: Amer Math Soc, 1978
[4] Coppel W A.Dichotomies in Stability Theory. Lecture Notes in Math 629. Berlin: Springer, 1978
[5] Chu J F, Zhao S F, Li M.Periodic shadowing of vector fields. Discrete Contin Dyn Syst, 2016, 36: 3623-3638
[6] Gan S B, Li M, Tikhomirov S B.Oriented shadowing property and -stability for vector fields. J Dynam Differential Equations, 2016, 28: 225-237
[7] Hayashi S.Connecting invariant manifolds and the solution of the C1-stability and -stability conjectures for flows. Ann of Math, 1997, 145: 81-137
[8] Han B.A flowbox theorem with uniform relative bound. Acta Mathematica Scientia, 2018, 38A(4): 625-630
[9] Lee M, Oh J, Wen X.Diffeomorphisms with a generalized Lipschitz shadowing property. Discrete Contin Dyn Syst, 2021, 41: 1913-1927
[10] Lee K, Sakai K.Structural stability of vector fields with shadowing. J Differential Equations, 2007, 232: 303-313
[11] Liao S T.An existence theorem for periodic orbits. Beijing Daxue Xuebao, 1979, 1: 1-20
[12] Liao S T.Obstruction sets I. Acta Math Sinica, 1980, 23: 411-453
[13] Liao S T.Qualitative Theory of Differentiable Dynamical Systems. Translated from the Chinese. With a preface by Min-de Cheng. Beijing: Science Press, distributed by Providence, RI: American Mathematical Society, 1996
[14] Moriyasu K, Sakai K, Sun W.C1-stably expansive flows. J Differential Equations, 2005, 213: 352-367
[15] Maizel’ A D.On stability of solutions of systems of differential equations. Ural Politehn Inst Tr, 1954, 51: 20-50
[16] Palmer K J, Pilyugin S Y, Tikhomirov S B.Lipschitz shadowing and structural stability of flows. J Diff Equ, 2012, 252: 1723-1747
[17] Pilyugin S Y.Shadowing in Dynamical Systems. Lecture Notes in Math 1706. Berlin: Springer, 1999
[18] Pilyugin S Y, Tikhomirov S B.Lipschitz shadowing implies structural stability. Nonlinearity, 2010, 23: 2509-2515
[19] Palmer K J.Shadowing in Dynamical Systems, Theory and Applications. Dordrecht: Kluwer, 2000
[20] Pliss V A.Bounded solutions of nonhomogeneous linear systems of differential equations//Problems of the Asymptotic Theory of Nonlinear Oscillations. Naukova Dumka, Kiev, 1997: 168-173
[21] Palmer K J.Exponential dichotomies and transversal homoclinic points. J Differential Equations, 1984, 55: 225-256
[22] Palmer K J.Exponential dichotomies and Fredholm operators. Proc Amer Math Soc, 1988, 104: 149-156
[23] Pilyugin S Y, Sakai K.Shadowing and Hyperoblicity. Lecture Notes in Math 2193. Berlin: Springer, 2017
[24] Pilyugin S Y, Tikhomirov S B.Lipschitz shadowing implies structural stability. Nonlinearity, 2010, 23: 2509-2515
[25] Pugh C, Shub M.The Ω-stability theorem for flows. Invent Math, 1970, 11: 150-158
[26] Plamenevskaya O B.Pseudo-orbit tracing property and limit shadowing property on a circle. Vestnik St Petersburg Univ Math, 1997, 30: 27-30
[27] Robinson C.Stability theorem and hyperbolicity in dynamical systems. Rocky Mountain J Math, 1977, 7: 425-437
[28] Robinson C.Stability theorems and hyperbolicity in dynamical systems. Rocky Mountain J Math, 1977, 7: 425-437
[29] Robinson C.Structural stability of vector fields. Ann of Math, 1974, 99: 154-175
[30] Sakai K.Pseudo orbit tracing property and strong transversality of diffeomorphisms of closed manifolds. Osaka J Math, 1994, 31: 373-386
[31] Sacker R J, Sell G R.Existence of dichotomies and invariant splittings for linear differential systems II. J Differential Equations, 1976, 22: 478-496
[32] Sacker R J, Sell G R.Existence of dichotomies and invariant splittings for linear differential systems III. J Differential Equations, 1976, 22: 497-522
[33] Todorov D.Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory. Discrete Contin Dyn Syst, 2013, 33: 4187-4205
[34] Wen L.Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity, Graduate Studies in Mathematics 173. American Mathematical Society, 2016
[35] Wen X, Gan S, Wen L.C1-stably shadowable chain components are hyperbolic. J Differential Equations, 2009, 246: 340-357
[36] Yuan G C, York J A.An open set of maps for which every point is absolutely non-shadowable. Proc Amer Math Soc, 2000, 128: 909-918
