Let τ be a null preserving point transformation on a finite measure space. Assuming τ is invertible, P. Ortega Salvador has recently obtained sufficient conditions for the almost everywhere convergence of the ergodic averages in with 1 < p < ∞, 1 < q < ∞. In this paper we obtain necessary and sufficient conditions for the almost everywhere convergence, without assuming that τ is invertible and only assuming that p ≠ ∞.
Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average converges almost everywhere to a function f* in , where (pq) and are assumed to be in the set . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified...
The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio concerning the Poisson boundary of random rational affinities.
For infinite measure preserving transformations with a compact regeneration property we establish a central limit theorem for visits to good sets of finite measure by points from Poissonian ensembles. This extends classical results about (noninteracting) infinite particle systems driven by Markov chains to the realm of systems driven by weakly dependent processes generated by certain measure preserving transformations.
We define a class of -actions, d ≥ 2, called product -actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing -action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic -action with Lebesgue component of multiplicity , where k is an arbitrary positive...