  We consider skew product dynamical systems
$f:\Theta\times\mathbb{R}\to\Theta\times\mathbb{R},
f(\theta,y)=(T\theta,f_\theta(y))$ with a (generalized) baker transformation
$T$ at the base and uniformly bounded increasing $C^3$ fibre maps $f_\theta$
with negative Schwarzian derivative. Under a partial hyperbolicity assumption
that ensures the existence of strong stable fibres for $f$ we prove that the
presence of these fibres restricts considerably the possible structures of
invariant measures - both topologically and measure theoretically, and that
this finally allows to provide a "thermodynamic formula" for the Hausdorff
dimension of set of those base points over which the dynamics are synchronized,
i.e. over which the global attractor consists of just one point.
