Ergodic Theory And Dynamical Systems
We define directional recurrence for infinite measure preserving Zd 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 Gδ 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.
Aimee S.A. Johnson and A. A. Şahin.
"Directional Recurrence For Infinite Measure Preserving Zᵈ Actions".
Ergodic Theory And Dynamical Systems.