Existence and non-existence of a finite invariant measure
This paper serves two distinct purposes. First, it collects into one place a list of seven conditions on a nonsingular transformation \(T\) on a finite measure space \(\{X, m\}\) which are known to be equivalent to the existence of a preserved measure \(\mu \sim m\). The authors provide a collection of techniques and results which are used to directly show their equivalence (Theorem 3.2). This section of the paper serves as a survey of prior results, rather than a new result in its own right. The second aim of the paper is the introduction of the notion of a ``strongly weakly wandering'' (sww) sequence, related to the existing notion of an ``exhaustive weakly wandering'' (eww) sequence. Whereas an (eww) sequence is associated to a set whose iterates under \(T\) given by the sequence are disjoint and cover \(X\), a (sww) sequence has no such covering requirement, albeit with a disjointness condition which more intricately depends on the arithmetic relation between terms of the sequence. The (sww) condition is shown to be equivalent to nonexistence of a preserved \(\mu \sim m\) (Propositions 4.5 and 4.6). The authors remark that few explicit examples exist of transformations with (eww) sequences, and that (sww) sequences are easier to construct, but do not include any such examples. Remarks indicate that a forthcoming monograph will contain further discussions, however.
scientific article; zbMATH DE number 6146574