The full diapason of convergence rates of Birkhoff averages for ergodic flows

TL;DR

This paper explores the full spectrum of convergence rates for Birkhoff averages in ergodic flows, demonstrating how different averaging functions can realize these rates, including the maximum and arbitrarily slow convergence.

Contribution

It extends Krengel's classical results by showing the range of convergence rates achievable through various averaging functions in ergodic flows and torus windings.

Findings

Range of convergence rates realized by averaging functions

Continuity of averaging functions in torus windings

Complement to Krengel's results on slow convergence

Abstract

For an ergodic flow, a range of rates of convergence of Birkhoff averages from the maximum rate to an arbitrarily slow rate is realized by choosing the averaging function. For torus windings, the continuity of the averaging functions is ensured. This complements Krengel's classical result on the slow rates of convergence of means for ergodic automorphisms.