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

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- by Ernest Fontich, Rafael de la Llave and Pau Martín PDF
- Trans. Amer. Math. Soc.
**358**(2006), 1317-1345

## 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.## References

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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
- Received by editor(s): April 9, 2003
- Received by editor(s) in revised form: May 11, 2004
- Published electronically: August 1, 2005
- © Copyright 2005 by the authors
- Journal: Trans. Amer. Math. Soc.
**358**(2006), 1317-1345 - MSC (2000): Primary 37D10, 37D25, 34D09, 70K45
- DOI: https://doi.org/10.1090/S0002-9947-05-03840-7
- MathSciNet review: 2187655