Superstatistics of Blaschke products

We consider a dynamics generated by families of maps whose invariant density depends on a parameter a and where a itself obeys a stochastic or periodic dynamics. For slowly varying a the long-term behavior of iterates is described by a suitable superposition of local invariant densities. We provide rigorous error estimates how good this approximation is. Our method generalizes the concept of superstatistics, a useful technique in nonequilibrium statistical mechanics, to maps. Our main example are Blaschke products, for which we provide rigorous bounds on the difference between Birkhoff density and the superstatistical composition.