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

I. V. Podvigin, V. V. Ryzhikov

TL;DR

For an ergodic flow $T_t$ and a function $f\in L_1(X,m)$, the paper shows that by selecting the averaging function in the form of a functional series one can realize the full diapason of convergence rates for Birkhoff averages $A(f,t,x)$, from the maximal rate to arbitrarily slow rates. It provides explicit constructions on a suspension flow over the dyadic odometer to realize both maximal and arbitrarily slow rates, and extends to general aperiodic flows via Rudolph's representation; it also demonstrates continuous realizations for torus windings by smoothing the averaging functions. The work complements Krengel's results on slow convergence for ergodic automorphisms and discusses distributions of Birkhoff average values and optimal weight distributions, highlighting how averaging schemes control convergence behavior. It thereby broadens the understanding of convergence dynamics in ergodic theory and offers a toolkit for engineering convergence rates in both discrete and continuous time settings.

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.

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

TL;DR

For an ergodic flow and a function , the paper shows that by selecting the averaging function in the form of a functional series one can realize the full diapason of convergence rates for Birkhoff averages , from the maximal rate to arbitrarily slow rates. It provides explicit constructions on a suspension flow over the dyadic odometer to realize both maximal and arbitrarily slow rates, and extends to general aperiodic flows via Rudolph's representation; it also demonstrates continuous realizations for torus windings by smoothing the averaging functions. The work complements Krengel's results on slow convergence for ergodic automorphisms and discusses distributions of Birkhoff average values and optimal weight distributions, highlighting how averaging schemes control convergence behavior. It thereby broadens the understanding of convergence dynamics in ergodic theory and offers a toolkit for engineering convergence rates in both discrete and continuous time settings.

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.
Paper Structure (12 sections, 92 equations, 6 figures)

This paper contains 12 sections, 92 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Convergence with maximum rate
  3. Slow convergence of Birkhoff averages for flow over a dyadic odometer
  4. Slow convergence of Birkhoff averages in the general case
  5. On the slow convergence of time averages for a torus winding and a continuous function
  6. Concluding remarks
  7. Введение
  8. Сходимоcть с максимальной скоростью
  9. Медленная сходимость средних Биркгофа для потока над диадическим одометром
  10. Медленные сходимости средних Биркгофа в общем случае
  11. О медленной сходимости временных средних для обмотки тора и непрерывной функции
  12. Заключительные замечания

Figures (6)

  • Figure 1: Self-similar graph of 2-odometer (left); Realization of $M_n, n=2$ using the flow over the 2-odometer (right)
  • Figure 2: Rudolph representation with parameters $h_n$ and $h_n+\varepsilon_n$
  • Figure 3: Parallelogram $R_n$ and definition of function $f_n$ on it
  • Figure 4: Самоподобный график 2-одометра (слева); реализация $M_n, n=2$ с помощью потока над 2-одометром (справа)
  • Figure 5: Представление Рудольфа с параметрами $h_n$ и $h_n+\varepsilon_n$
  • ...and 1 more figures