Superstatistics of Blaschke products

TL;DR

This paper extends superstatistics to dynamical maps, specifically Blaschke products, providing rigorous bounds on the approximation of long-term densities when parameters vary slowly.

Contribution

It introduces a rigorous framework for applying superstatistics to maps, with explicit error estimates and bounds for Blaschke products.

Findings

Superstatistical approximation closely matches Birkhoff densities for Blaschke products.

Provided explicit error bounds for the superstatistical approximation.

Demonstrated the applicability of superstatistics to deterministic maps.

Abstract

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.