2023/04/03 by Bonanno, Claudio, Schindler, Tanja I.
#11K50 #37A25 #37A40 #60F15 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2304.01132
We consider a conservative ergodic measure-preserving transformation T of the measure space (X,B,μ) with μ a σ-finite measure and μ(X)=∞. Given an observable g:X→ ℝ, it is well known from results by Aaronson that in general the asymptotic behaviour of the Birkhoff sums SNg(x):= ∑j=1N (g∘ Tj-1)(x) strongly depends on the point x∈ X, and that there exists no sequence (dN) for which SNg(x)/dN → 1 for μ-almost every x∈ X. In this paper we consider the case g\not∈ L1(X,μ) assuming that there exists E\inB with μ(E)<∞ and ∫E g dμ=∞ and continue the investigation initiated in previous work by the authors. We show that for transformations T with strong mixing assumptions for the induced map on a finite measure set, the almost sure asymptotic behaviour of SNg(x) for an unbounded observable g may be obtained using two methods, adding a number of summands depending on x to SNg and trimming. The obtained sums are then asymptotic to a scalar multiple of N. The results are applied to a couple of non-regular continued fraction algorithms, the backward (or Rényi type) continued fraction and the even-integer continued fraction algorithms, to obtain the almost sure asymptotic behaviour of the sums of the digits of the algorithms.