Generalized Fiber Contraction Mapping Principle

Abstract

We prove a generalized non-stationary version of the fiber contraction mapping theorem. It was originally used by Hirsch and Pugh, in 1970, to prove that the stable foliation of a C² Anosov diffeomorphism of a surface is C¹. Our generalized principle was recently used by the first author to prove an analogous regularity result for stable foliations of non-stationary systems. The result is stated in a general setting so that it may be used in future dynamical results in the random and non-stationary settings, especially for graph transform arguments.