# American Institute of Mathematical Sciences

May  2012, 32(5): 1537-1555. doi: 10.3934/dcds.2012.32.1537

## Stable manifolds with optimal regularity for difference equations

 1 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa 2 Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  December 2010 Revised  April 2011 Published  January 2012

We obtain stable invariant manifolds with optimal $C^k$ regularity for a nonautonomous dynamics with discrete time. The dynamics is obtained from a sufficiently small perturbation of a nonuniform exponential dichotomy, which includes the notion of (uniform) exponential dichotomy as a very special case. We emphasize that we do not require the dynamics to be of class $C^{k+\epsilon}$, in strong contrast to former results in the context of nonuniform hyperbolicity. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, our method also allows linear perturbations, and thus the results readily apply to the robustness problem of nonuniform exponential dichotomies.
Citation: Luis Barreira, Claudia Valls. Stable manifolds with optimal regularity for difference equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1537-1555. doi: 10.3934/dcds.2012.32.1537
##### References:
 [1] L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,", Encyclopedia of Math. and Its Appl., 115 (2007).   Google Scholar [2] L. Barreira and C. Valls, Existence of stable manifolds for nonuniformly hyperbolic $C^1$ dynamics,, Discrete Contin. Dyn. Syst., 16 (2006), 307.  doi: 10.3934/dcds.2006.16.307.  Google Scholar [3] L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math., 1926 (2008).   Google Scholar [4] C. Chicone, "Ordinary Differential Equations with Applications,", Second edition, 34 (2006).   Google Scholar [5] A. Fathi, M. Herman and J.-C. Yoccoz, A proof of Pesin's stable manifold theorem,, in, 1007 (1983), 177.   Google Scholar [6] R. Mañé, Lyapounov exponents and stable manifolds for compact transformations,, in, 1007 (1983), 522.   Google Scholar [7] V. Oseledec, A multiplicative ergodic theorem. Charactersitc Ljapunov, exponents of dynamical systems,, Trudy Moskov. Mat. Obšč., 19 (1968), 179.   Google Scholar [8] Ja. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 1332.   Google Scholar [9] Ja. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory,, Uspehi Mat. Nauk, 32 (1977), 55.   Google Scholar [10] Ja. Pesin, Geodesic flows on closed Riemannian manifolds without focal points,, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1252.   Google Scholar [11] C. Pugh and M. Shub, Ergodic attractors,, Trans. Amer. Math. Soc., 312 (1989), 1.  doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar [12] D. Ruelle, Ergodic theory of differentiable dynamical systems,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 27.   Google Scholar [13] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space,, Ann. of Math. (2), 115 (1982), 243.  doi: 10.2307/1971392.  Google Scholar

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##### References:
 [1] L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,", Encyclopedia of Math. and Its Appl., 115 (2007).   Google Scholar [2] L. Barreira and C. Valls, Existence of stable manifolds for nonuniformly hyperbolic $C^1$ dynamics,, Discrete Contin. Dyn. Syst., 16 (2006), 307.  doi: 10.3934/dcds.2006.16.307.  Google Scholar [3] L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math., 1926 (2008).   Google Scholar [4] C. Chicone, "Ordinary Differential Equations with Applications,", Second edition, 34 (2006).   Google Scholar [5] A. Fathi, M. Herman and J.-C. Yoccoz, A proof of Pesin's stable manifold theorem,, in, 1007 (1983), 177.   Google Scholar [6] R. Mañé, Lyapounov exponents and stable manifolds for compact transformations,, in, 1007 (1983), 522.   Google Scholar [7] V. Oseledec, A multiplicative ergodic theorem. Charactersitc Ljapunov, exponents of dynamical systems,, Trudy Moskov. Mat. Obšč., 19 (1968), 179.   Google Scholar [8] Ja. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 1332.   Google Scholar [9] Ja. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory,, Uspehi Mat. Nauk, 32 (1977), 55.   Google Scholar [10] Ja. Pesin, Geodesic flows on closed Riemannian manifolds without focal points,, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1252.   Google Scholar [11] C. Pugh and M. Shub, Ergodic attractors,, Trans. Amer. Math. Soc., 312 (1989), 1.  doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar [12] D. Ruelle, Ergodic theory of differentiable dynamical systems,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 27.   Google Scholar [13] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space,, Ann. of Math. (2), 115 (1982), 243.  doi: 10.2307/1971392.  Google Scholar
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