Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing

TL;DR

This paper investigates bifurcations of invariant graphs and the Hausdorff dimension in skew product systems driven by hyperbolic maps with multiplicative forcing, revealing how dimensions change with parameters.

Contribution

It characterizes the parameter dependence of invariant graph bifurcations and the Hausdorff dimension of the attractor using thermodynamic formalism in hyperbolic-driven systems.

Findings

Existence of coexisting invariant graphs for certain parameters.

Dimension of the global attractor drops from 3 to lower values at a critical parameter.

Dimension varies strictly decreasing beyond the critical parameter t_c.

Abstract

We study bifurcations of invariant graphs in skew product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with one-dimensional concave fibre maps under multiplicative forcing when the forcing is scaled by a parameter r=e^{-t}. For a range of parameters two invariant graphs (a trivial and a non-trivial one) coexist, and we use thermodynamic formalism to characterize the parameter dependence of the Hausdorff and packing dimension of the set of points where both graphs coincide. As a corollary we characterize the parameter dependence of the dimension of the global attractor A_t: Hausdorff and packing dimension have a common value dim(A_t), and there is a critical parameter t_c determined by the SRB measure of T^{-1} such that dim(A_t)=3 for t < t_c and t --> dim(A_t) is strictly decreasing for t_c < t < t_{max}.