Existence of stable manifolds for nonuniformly hyperbolic C1 dynamics

Abstract

The existence of stable manifolds for nonuniformly hyperbolic trajectories is well known in the case of C1+α dynamics, as proven by Pesin in the late 1970's. On the other hand, Pugh constructed a C1 diffeomorphism that is not of class C1+α for any a, and for which there exists no stable manifold. The C1+α 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 C1, by providing a condition which is weaker than the C1+α 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+α nonuniformly hyperbolic dynamics. In particular, the optimal C1 smoothness of the invariant manifolds is obtained by constructing an invariant family of cones.