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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems


Authors: Ernest Fontich, Rafael de la Llave and Pau Martín
Journal: Trans. Amer. Math. Soc. 358 (2006), 1317-1345
MSC (2000): Primary 37D10, 37D25, 34D09, 70K45
Published electronically: August 1, 2005
MathSciNet review: 2187655
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Abstract: Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to ``slow manifolds'', which characterize the asymptotic convergence.

Let $\{x_i\}_{i \in \mathbb{N} }$ be a regular orbit of a $C^2$ dynamical system $f$. Let $S$ be a subset of its Lyapunov exponents. Assume that all the Lyapunov exponents in $S$ are negative and that the sums of Lyapunov exponents in $S$ do not agree with any Lyapunov exponent in the complement of $S.$ Denote by $E^S_{x_i}$ the linear spaces spanned by the spaces associated to the Lyapunov exponents in $S.$ We show that there are smooth manifolds $W^S_{x_i}$ such that $f(W^S_{x_i}) \subset W^S_{x_{i+1}}$ and $T_{x_i} W^S_{x_i} = E^S_{x_i}$. We establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non-resonance conditions. These systems of invariant manifolds are not, in general, a foliation.


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Additional Information

Ernest Fontich
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
Email: fontich@mat.ub.es

Rafael de la Llave
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712-1082
Email: llave@math.utexas.edu

Pau Martín
Affiliation: Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Ed-C3, Jordi Girona, 1-3, 08034 Barcelona, Spain
Email: martin@ma4.upc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03840-7
PII: S 0002-9947(05)03840-7
Keywords: Lyapunov exponents, invariant manifolds, resonaces, normal forms
Received by editor(s): April 9, 2003
Received by editor(s) in revised form: May 11, 2004
Published electronically: August 1, 2005
Article copyright: © Copyright 2005 by the authors