Today I saw a paper on arxiv entitled with “semiuniform ergodic theorem”. This is the first time I saw such a theorem and I want to take a note about it.

Let’s start with , a homeomorphism on a compact metric space. Let be the collection of -invariant probability measures on . Let be a continuous function. Birkhoff ergodic theorem states that the time-average for almost every , where is a almost every defined, measurable function.

A special case is that is a singleton. Such a map is called uniquely erogdic. In this case the time-average converges uniformly to a constant for every .

The uniform ergodic theorem concerns an intermediate case, that is a singleton (say also ), and states that the time-average of converges uniformly to a constant for every .

For the general case, is a compact interval, say . Let’s show that the time average will fall close to this interval uniformly on .

Proof. We will derive a contradiction by assuming the contrary that, there exists , and such that for every . Then consider the sequence . Passing to a subsequence if necessary, we can assume that it converges to some , which will force and contradict the choice of . Q.E.D.

Then let’s consider a subadditive sequence . Denote and . Then Semiuniform Erogdic Theorem concerns one-side estimate similar to UET. It states that if , then there exist and such that for every , and every .

Proof. We will derive a contradiction by assuming the contrary that, for each , there exist and such that for every . Then consider the sequence . Passing to a subsequence if necessary, we can assume that it converges to some . Then we use a common trick to show and hence contradicts the choice of . Note that it suffices to show for each . From now on let’s fix .

Let and decompose for some . So . Summing over and divide both sides by , we get

. Now passing we get and hence . Q.E.D.

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