‎This paper investigates the geometrical structures of invariant graphs of skew product systems of the form $F‎ : ‎\Theta \times \I \to \Theta \times \I‎ , ‎(\theta,y)\mapsto (S\theta,\f(y))$ 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 \cite{KO2}‎, ‎the authors studied skew product systems driven by a generalized baker map $S:\Tt\to\Tt$ with the restrictive assumption that $\f$ depend on $\theta=\2$ 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 drive. As an application of this construction we prove that, when S is an Anosov diffeomorphism, a pinching set is a union of global unstable fibers for S. ‎

Keywords