Stable manifolds with optimal regularity for difference equations

Luis Barreira; Claudia Valls

doi:10.3934/dcds.2012.32.1537

Article Contents

2012, Volume 32, Issue 5: 1537-1555. Doi: 10.3934/dcds.2012.32.1537

This issue Previous Article Dimensional reduction for supremal functionals Next Article Periodic and subharmonic solutions for duffing equation with a singularity

Stable manifolds with optimal regularity for difference equations

Luis Barreira^1, and
Claudia Valls^2,

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

Received: November 30, 2010

Revised: March 31, 2011

Published: April 2012

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Difference equations,
stable manifolds.

Mathematics Subject Classification: Primary: 37D10, 37D25.

Citation:

Related Papers

Cited by

References

[1]	L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Math. and Its Appl., 115, Cambridge Univ. Press, Cambridge, 2007.
[2]	L. Barreira and C. Valls, Existence of stable manifolds for nonuniformly hyperbolic $C^1$ dynamics, Discrete Contin. Dyn. Syst., 16 (2006), 307-327.doi: 10.3934/dcds.2006.16.307.
[3]	L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math., 1926, Springer, Berlin, 2008.
[4]	C. Chicone, "Ordinary Differential Equations with Applications," Second edition, Texts in Applied Mathematics, 34, Springer, New York, 2006.
[5]	A. Fathi, M. Herman and J.-C. Yoccoz, A proof of Pesin's stable manifold theorem, in "Geometric Dynamics" (ed. J. Palis, Rio de Janeiro, 1981), Lect. Notes. in Math., 1007, Springer, Berlin, (1983), 177-215.
[6]	R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in "Geometric Dynamics" (ed. J. Palis, Rio de Janeiro, 1981), Lect. Notes in Math., 1007, Springer, Berlin, (1983), 522-577.
[7]	V. Oseledec, A multiplicative ergodic theorem. Charactersitc Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.
[8]	Ja. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 1332-1379.
[9]	Ja. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-114, 287.
[10]	Ja. Pesin, Geodesic flows on closed Riemannian manifolds without focal points, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1252-1288, 1447.
[11]	C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54.doi: 10.1090/S0002-9947-1989-0983869-1.
[12]	D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 27-58.
[13]	D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290.doi: 10.2307/1971392.