Pointwise convergence of Birkhoff averages for global observables

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.

arXiv commentFinal version. 33 pages, 10 figures

References in corpus (10)

• Lévy walks
• On infinite-volume mixing
• Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems
• Infinite mixing for one-dimensional maps with an indifferent fixed point
• Uniformly expanding Markov maps of the real line: exactness and infinite mixing
• Transport properties of Lévy walks: an analysis in terms of multistate processes
• Strong laws of large numbers for intermediately trimmed sums of i.i.d. random variables with infinite mean
• Exactness, K-property and infinite mixing
• Dynamical Borel-Cantelli lemmas and rates of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems
• Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations

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• A Kingman-like Theorem
• Almost sure limit theorems with applications to non-regular continued fraction algorithms