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

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.