Invariant Graphs for Chaotically Driven Maps

TL;DR

This paper explores the structure of invariant graphs in skew product systems driven by hyperbolic maps, focusing on the pinching set and relaxing previous restrictive assumptions to generalize the analysis.

Contribution

It generalizes the classification of invariant graphs by removing previous restrictions on fibre map dependence, enabling broader applicability.

Findings

Describes the structure of invariant graphs and pinching sets.

Provides a fibre-wise conjugation to simplify the system.

Extends previous results to more general hyperbolic base maps.

Abstract

This paper investigates the geometrical structures of invariant graphs of skew product systems of the form $F : \Theta \times I \to \Theta \times I , (\theta,y)\mapsto (S\theta,f_\theta(y))$ driven by a hyperbolic base map $S : \Theta \to \Theta$ (e.g. a baker map or an Anosov surface diffeomorphism) and with monotone increasing fibre maps $(f_{\theta})_{\theta \in \Theta}$ having negative Schwarzian derivatives. We recall a classification, with respect to the number and to the Lyapunov exponents of invariant graphs, for this class of systems. Our major goal here is to describe the structure of invariant graphs and study the properties of the pinching set, the set of points where the values of all of the invariant graphs coincide. In arXiv:1610.10010, the authors studied skew product systems driven by a generalized Baker map $S$ with the restrictive assumption that $f_\theta$ depends on $\theta=(\xi,x)$ only through the stable coordinate $x$ of $\theta$. Our aim is to relax this assumption and construct a fibre-wise conjugation between the original system and a new system for which the fibre maps depend only on the stable coordinate of the derive.