41 research outputs found
Under recurrence in the Khintchine recurrence theorem
The Khintchine recurrence theorem asserts that on a measure preserving
system, for every set and , we have for infinitely many . We show that there
are systems having under-recurrent sets , in the sense that the inequality
holds for every . In
particular, all ergodic systems of positive entropy have under-recurrent sets.
On the other hand, answering a question of V.~Bergelson, we show that not all
mixing systems have under-recurrent sets. We also study variants of these
problems where the previous strict inequality is reversed, and deduce that
under-recurrence is a much more rare phenomenon than over-recurrence. Finally,
we study related problems pertaining to multiple recurrence and derive some
interesting combinatorial consequences.Comment: 18 pages. Referee's comments incorporated. To appear in the Israel
Journal of Mathematic
On two recurrence problems
We review some aspects of recurrence in topological dynamics and focus on two
open problems. The first is an old one concerning the relation between Poincare
and Birkhoff recurrence; the second, due to Boshernitzan, is about moving
recurrence. We provide a partial answer to a topological version of the moving
recurrence problem