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arXiv · 1610.10010

Chaotically driven sigmoidal maps

Gerhard Keller·Atsuya Otani
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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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Gerhard Keller, Atsuya Otani. 2016-10-31. Chaotically driven sigmoidal maps. https://doi.org/10.1142/s0219493718500090

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