Invariant graphs for chaotically driven maps

Abstract

This paper investigates the geometrical structures of invariant graphs of skew product systems of the form driven by a hyperbolic base map (e.g. a baker map or an Anosov surface diffeomorphism) and with monotone increasing fibre maps 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 Keller and Otani (2018 Stoch. Dyn. 18 1850009), the authors studied skew product systems driven by a generalized baker map with the restrictive assumption that depend on only through the stable coordinate x of θ. 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 drive. As an application of this construction we prove that, when S is an Anosov diffeomorphism, the pinching set is a union of global unstable fibres for S.