Generalized Fiber Contraction Mapping Principle

TL;DR

This paper introduces a generalized fiber contraction mapping theorem applicable to non-stationary systems, enabling advanced analysis of stable foliations in dynamical systems with potential applications in random and non-stationary contexts.

Contribution

It extends the classical fiber contraction mapping theorem to non-stationary settings, broadening its applicability in dynamical systems theory.

Findings

Proves a generalized non-stationary fiber contraction mapping theorem.

Demonstrates the theorem's use in analyzing stable foliations of non-stationary systems.

Provides a framework for future research in random and non-stationary dynamical systems.

Abstract

We prove a generalized non-stationary version of the fiber contraction mapping theorem. It was originally used in [HirschPugh70] to prove that the stable foliation of a $C^2$ Anosov diffeomorphism of a surface is $C^1$. Our generalized principle is used in [Luna24], where an analogous regularity result for stable foliations of non-stationary systems is proved. 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.