On the Regularity of Invariant Foliations

We show that the stable invariant foliation of codimension 1 near a zero-dimensional hyperbolic set of a $C^{\beta}$ map with $\beta>1$ is $C^{1+\varepsilon}$ with some $\varepsilon>0$. The result is applied to the restriction of higher regularity maps to normally hyperbolic manifolds. An application to the theory of the Newhouse phenomenon is discussed.