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A function with locally uncountable rotation set. (English) Zbl 0963.28013

Summary: The rotain set \(\Gamma\) of a Lebesgue measurable real valued function on the circle is the set of \(\alpha \in \mathbb R\) for wich \(\frac 1{n+1}\sum^n_{k=0}f(x+k\alpha)\) converges as \(n\to \infty\) for almost every \(x\). Z. Buczolich [Ergodic Theory Dyn. Syst. 16, No. 6, 1185-1196 (1996; Zbl 0869.28008)] has constructed a non-integrable function whose rotation set contains at least countably many irrational rotations. In this paper we construct a function whose rotation set has uncountable intersection with each non-empty open subset of \(\mathbb R\).

MSC:

28D05 Measure-preserving transformations

Citations:

Zbl 0869.28008
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