Chaotically driven sigmoidal maps
  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. 
Gerhard Keller and Atsuya Otani
math.DS