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 regularitymaps to normally hyperbolic manifolds. An application to the theory of the Newhouse phenomenon is discussed.