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 ∞, which we generically call global observables. Our main theorem applies to general systems but requires a 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.

Pointwise convergence of Birkhoff averages for global observables / Lenci, Marco; Munday, Sara. - In: CHAOS. - ISSN 1054-1500. - STAMPA. - 28:8(2018), pp. 083111.083111-1-083111.083111-16. [10.1063/1.5036652]

Pointwise convergence of Birkhoff averages for global observables

Lenci, Marco
;
2018

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 ∞, which we generically call global observables. Our main theorem applies to general systems but requires a 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.
2018
Pointwise convergence of Birkhoff averages for global observables / Lenci, Marco; Munday, Sara. - In: CHAOS. - ISSN 1054-1500. - STAMPA. - 28:8(2018), pp. 083111.083111-1-083111.083111-16. [10.1063/1.5036652]
Lenci, Marco; Munday, Sara
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/649076
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