Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

2021/04/21 by Bonanno, Claudio, Schindler, Tanja I.
#37A25 #37A40 #60F15 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2104.10458

Abstract

We consider a conservative ergodic measure-preserving transformation T of a σ-finite measure space (X,B,μ) with μ(X)=∞. Given an observable f:X→ ℝ we study the almost sure asymptotic behaviour of the Birkhoff sums S_Nf(x) := ∑_j=1^N (f∘ T^j-1)(x). In infinite ergodic theory it is well known that the asymptotic behaviour of S_Nf(x) strongly depends on the point x∈ X, and if f∈ L¹(X,μ), then there exists no real valued sequence (b(N)) such that lim_N→∞ S_Nf(x)/b(N)=1 almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence (α(N)) and m\colon X× ℕ→ℕ such that for f∈ L¹(X,μ) we have lim_N→∞ S_N+m(x,N)f(x)/α(N)=1 for μ-a.e. x∈ X. Moreover if f\not∈ L¹(X,μ) we give conditions on the induced observable such that there exists a sequence (G(N)) depending on f, for which lim_N→∞ S_Nf(x)/G(N)=1 holds for μ-a.e. x∈ X.

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