Pointwise convergence of Birkhoff averages for global observables

TL;DR

This paper extends Birkhoff's ergodic theorem to certain global observables in infinite-measure-preserving systems, providing conditions for pointwise convergence of averages beyond integrable functions.

Contribution

It introduces a version of Birkhoff's theorem for global observables in infinite ergodic systems under an approximate partial averaging hypothesis.

Findings

Main theorem applies to general systems with specific observables.

Counterexamples and simulations explore optimal observable classes.

The result broadens understanding of pointwise averages in infinite ergodic theory.

Abstract

It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost everywhere zero. Nor does a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In this paper we give a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of $L^\infty$, which we generically call global observables. Our main theorem applies to general systems but requires an hypothesis of "approximate partial averaging" on the observables. The idea behind the result, however, applies to more general situations, as we show with an example. Finally, by means of counterexamples and numerical simulations, we discuss the question of finding the optimal class of observables for which a Birkhoff theorem holds for infinite-measure-preserving systems.