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 behaviour 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 is Blaschke products, for which we provide rigorous error estimates on the difference between Birkhoff density and the superstatistical approximation.