Smooth stable manifolds in Rn

In this chapter we start the study of the regularity of the Lipschitz manifolds constructed in Chapter 4. We only consider stable manifolds. As in Section 4.5, the theory for unstable manifolds is analogous, and the proofs can be readily obtained by reversing the time. We only consider in this chapter the case of finite-dimensional spaces. This is due to the method of proof of the smoothness of the invariant manifolds, which uses in a decisive manner the compactness of the closed unit ball in Rn (in the proof of Lemma 5.11). The proof is based on the construction of an invariant family of cones, in a similar manner to that in the classical hyperbolic theory, although now using an appropriate family of Lyapunov norms. The family of cones allows us to obtain an invariant distribution which coincides with the tangent bundle of the invariant manifold. This also allows us to discuss the continuity of the distribution, and thus the continuity of the tangent spaces, that corresponds to the smoothness of the invariant manifold. We note that we deal directly with the semiflows instead of first considering time-1 maps as it is sometimes customary in hyperbolic dynamics. The infinite-dimensional case is treated in Chapter 6 with an entirely different approach, although at the expense of requiring more regularity for the vector field. The material in this chapter is taken from [6] (for Sections 5.1–5.4) and [5] (for Sections 5.5–5.6), although now considering the general case when the stable and unstable subspaces may depend on the time t.