Sequences of integers and ergodic transformations
A class of infinite measure preserving transformations is constructed related to direct sum decompositions of the natural numbers \({\mathbb{N}}\). Specifically, let \({\mathbb{V}}\) and \({\mathbb{W}}\) be infinite subsets of \({\mathbb{N}}\) such that each \(n\in {\mathbb{N}}\) is a unique sum \(n=v+w,\) with \(v\in {\mathbb{V}}\) and \(w\in {\mathbb{W}}.\) Then an ergodic infinite measure preserving transformation T is constructed with a special set A and a distinguished point x, such that: \(\mu (A)=1;\) the set A is exhaustive, weakly wandering for T under the sequence \({\mathbb{W}}\); and the return times of the point x to A is the sequence \({\mathbb{V}}\). It then follows that for almost every point \(a\in A,\) the return times of a to A, \({\mathbb{R}}(a)\) is a direct complement of \({\mathbb{W}}\) in the integers \({\mathbb{Z}}\), i.e. every \(z\in {\mathbb{Z}}\) is a unique sum, \(z=r\oplus w\) for \(r\in {\mathbb{R}}(a)\) and \(w\in {\mathbb{W}}.\)
ergodic infinite measure preserving transformation
scientific article; zbMATH DE number 4099608