Introduction

We study the laminations by stable and unstable manifolds associated with a C¹⁺ hyperbolic diffeomorphism. We show that the holonomies between the 1-dimensional leaves are C^1+α, for some 0<α≤1, and that the holonomies vary Hölder continuously with respect to the domain and target leaves. Hence, the laminations by stable and unstable manifolds are C¹⁺ foliated. This result is very useful in a number of contexts and it is used in all of the following chapters of the book. In general terms, it allows one to reduce many questions about 2-dimensional dynamics to questions about 1-dimensional dynamics.