Symmetric Birkhoff sums in infinite ergodic theory

Abstract

We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.

References in corpus (2)

• On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations
• Distribution of orbits in ${\mathbb{R}}^2$ of a finitely generated group of $\SL(2,\R)$

Cited by in corpus (2)

• Dynamical Borel-Cantelli lemmas and rates of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems
• Almost sure limit theorems with applications to non-regular continued fraction algorithms