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.

Chaotically driven sigmoidal maps - Keller, Gerhard; Otani, Atsuya (Stochastics and Dynamics, 2017)
We consider skew product dynamical systems f:Θ×ℝ→Θ×ℝ,f(휃,y)=(T휃,f휃(y))" role="presentation">f:Θ×ℝ→Θ×ℝ,f(𝜃,y)=(T𝜃,f𝜃(y)) with a (generalized) baker transformation T" role="presentation">T at the base and uniformly bounded increasing C3" role="presentation">C3 fibre maps f휃" role="presentation">f𝜃 with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for f" role="presentation">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 poin
Keywords: Skew product; negative Schwarzian; attractor; Hausdorff dimension