Directional recurrence for infinite measure preserving actions

Abstract

We define directional recurrence for infinite measure preserving $\mathbb{Z}^{d}$ actions both intrinsically and via the unit suspension flow and prove that the two definitions are equivalent. We study the structure of the set of recurrent directions and show it is always a $G_{{\it\delta}}$ set. We construct an example of a recurrent action with no recurrent directions, answering a question posed in a 2007 paper of Daniel J. Rudolph. We also show by example that it is possible for a recurrent action to not be recurrent in an irrational direction even if all its sub-actions are recurrent.

Locations

• Ergodic Theory and Dynamical Systems - View
• arXiv (Cornell University) - View - PDF
• Swarthmore College Works (Swarthmore College Libraries) - View - PDF
• DataCite API - View