  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}.
