The authors study bifurcations of global attractors in a family \((T_t)_{t\in\mathbb R}\) of dynamical systems of skew-product type which have one-dimensional concave fiber maps of particular multiplicative type \[ T_t:\Theta\times\mathbb R_\geq\to\Theta\times\mathbb R_\geq,\quad T_t(\theta,x):=(T\theta,d^{-t}g(\theta)h(x)), \] where \(g\) is bounded measurable and \(h\in C^1(\mathbb R_\geq;\mathbb R)\) is strictly concave, strictly increasing, and satisfies \(h(0)=0\), \(h'(0)=1\). Because of the particular shape of the fiber maps, for certain parameters there are two coexisting invariant graphs, the trivial one at \(x=0\) and a non-trivial one, that bound the global attractor \(\mathcal A_t\) of \(T_t\). The authors study the set of points on the base \(\Theta\) where both graphs coincide and explain how its dimension changes with the parameter \(t\). To do so, they first characterize the corresponding level sets in terms of Birkhoff averages which are closely related to fiberwise local backward Lyapunov exponents. These first investigations are completely general and do not depend on the base map \(T: \Theta\to\Theta\). See also G. Keller [Fundam. Math. 151, No. 2, 139–148 (1996; Zbl 0899.58033)] for related problems.
In the particular case in which the base map \(T\) is a topologically mixing \(C^2\) Anosov diffeomorphism and \(g\) is a Hölder continuous function (not cohomologous to a constant), using the thermodynamic formalism, the authors derive formulas for the Hausdorff dimension and the packing dimension of the level sets and of \(\mathcal A_t\). Here \(t\) ranges in the interval \((\gamma_{\min},\gamma_{\max})\) whose end points correspond to the minimal/maximal averages of the potential \(\log g\) averaged with respect to an ergodic probability in the base \(\Theta\). The average with respect to the SRB measure for the map \(T^{-1}\) defines a critical parameter \(\gamma_c^-\) for which the dimension of \(\mathcal A_t\) changes from being equal to \(3\) in \((\gamma_{\text{min}},\gamma_c^-]\) to being strictly decreasing in \([\gamma_c^-,\gamma_{\text{max}})\).