Chaotically driven sigmoidal maps

TL;DR

This paper investigates skew product dynamical systems with baker transformation bases and bounded fibre maps, revealing how stable fibres influence invariant measures and enabling a formula for the Hausdorff dimension of synchronized base points.

Contribution

It establishes the restrictive role of stable fibres on invariant measures and derives a thermodynamic formula for the Hausdorff dimension of synchronized base points.

Findings

Stable fibres limit the complexity of invariant measures.

A thermodynamic formula for Hausdorff dimension of synchronized points is derived.

Presence of stable fibres constrains the structure of the global attractor.

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.