On the multiple recurrence properties for disjoint systems

Abstract

We consider a mutually disjoint family of measure preserving transformations T1, …, Tk on a probability space \((X,{\cal B},\mu )\). We obtain the multiple recurrence property of T1, …, Tk and this result is utilized to derive multiple recurrence of Poincaré type in metric spaces. We also present the multiple recurrence property of Khintchine type. Further, we study multiple ergodic averages of disjoint systems and we show that T1, …, Tk are uniformly jointly ergodic if each Ti is ergodic.