Over Recurrence for Mixing Transformations

TL;DR

This paper investigates the existence of over-recurrent sets in mixing transformations, proving their universal presence in strong mixing cases and constructing examples without under-recurrent sets, thus addressing a question by V. Bergelson.

Contribution

It establishes that all invertible strong mixing transformations have over-recurrent sets and provides a method to construct strong mixing transformations lacking under-recurrent sets.

Findings

Strong mixing transformations always have over-recurrent sets.

Explicit construction of strong mixing transformations with no under-recurrent sets.

Existence of $ ext{ extepsilon}$-over-recurrent sets in ergodic transformations.

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 $\epsilon$-over-recurrence and show that given $\epsilon > 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $\epsilon$-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.