Chaotically driven sigmoidal maps

Abstract

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.

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