Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing - Keller, Gerhard; Otani, Atsuya (Dynamical Systems-An International Journal, 2013)
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.