June  2006, 16(2): 307-327. doi: 10.3934/dcds.2006.16.307

Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics

1. 

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

2. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  June 2005 Revised  July 2005 Published  July 2006

The existence of stable manifolds for nonuniformly hyperbolic trajectories is well know in the case of $C^{1+\alpha}$ dynamics, as proven by Pesin in the late 1970's. On the other hand, Pugh constructed a $C^1$ diffeomorphism that is not of class $C^{1+\alpha}$ for any $\alpha$ and for which there exists no stable manifold. The $C^{1+\alpha}$ hypothesis appears to be crucial in some parts of smooth ergodic theory, such as for the absolute continuity property and thus in the study of the ergodic properties of the dynamics. Nevertheless, we establish the existence of invariant stable manifolds for nonuniformly hyperbolic trajectories of a large family of maps of class at most $C^1$, by providing a condition which is weaker than the $C^{1+\alpha}$ hypothesis but which is still sufficient to establish a stable manifold theorem. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We note that our proof of the stable manifold theorem is new even in the case of $C^{1+\alpha}$ nonuniformly hyperbolic dynamics. In particular, the optimal $C^1$ smoothness of the invariant manifolds is obtained by constructing an invariant family of cones.
Citation: Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307
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