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arXiv · 1701.04345

Over Recurrence for Mixing Transformations

Terrence Adams
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Abstract

We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Bergelson. We define $ε$-over-recurrence and show that given $ε> 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $ε$-over-recurrent sets of arbitrarily small measure. Discrete spectrum transformations and rotations do not have over-recurrent sets, but we construct a weak mixing rigid transformation with strictly over-recurrent sets.

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Terrence Adams. 2019-02-28. Over Recurrence for Mixing Transformations. https://arxiv.org/abs/1701.04345

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