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

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 : Hausdorff and packing dimension have a common value , and there is a critical parameter γ− c determined by the SRB measure of T −1 such that for t⩽γ− c and is strictly decreasing for t∈[γ− c , γmax ).