  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.
