  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.
