A measure-preserving transformation of a (finite) measure space is called ergodic if whenever satisfies we have or . To aid intuition, it is useful to keep in mind the ergodic theorem, which roughly states that ergodicity equals equidistribution.
Following a discussion today, I’m concerned here with the ergodicity of for . This is an entertaining topic, and I thank Rudi Mrazovic for bringing it to my attention. Here’s an amusing example: While being ergodic does not generally imply that is ergodic, being ergodic does imply that is ergodic.
Let’s start with a quite simple observation.
Lemma 1 If and is ergodic then is ergodic.
Proof: Suppose . Then , so or .
What is the picture of being ergodic and not being ergodic? Certainly one picture to imagine is a partition with and , or in other words a factor . Possibly surprisingly, this is the whole story.
Theorem 2 If is ergodic and is not ergodic then for some divisor of there exists a partition of as where and .
Proof: Since is not ergodic there exists a set with and and . For any subset let consist of those such that
for some . (Note that because iff , the sentence “” makes sense for .) Note that , so or . Since , we must have for some .
Let be the smallest positive period of . Note that iff , contradicting , or , contradicting , so . Let
(Again note that makes sense for .) Then unless , i.e., iff is divisible by , in which case . Thus we have the desired partition
This finishes the proof.
Note as a corollary that “only the primes matter” insofar as which make ergodic: if and is not ergodic then for some prime , is not ergodic. Combining this with the initial lemma, which shows that if is ergodic then is ergodic for every , we have thus proved one half of the following theorem.
Theorem 3 Given a set , there exists a measure-preserving system such that if and only if is empty or the set of not divisible by any , for some set of primes .
It remains only to construct a measure-preserving system for a given set of primes. For finite this can be achieved even with finite spaces. In general it suffices to look at .