On the multiple recurrence properties for disjoint systems

TL;DR

This paper investigates multiple recurrence properties for disjoint measure-preserving systems, deriving new results on recurrence in metric spaces and establishing conditions for uniform joint ergodicity.

Contribution

It introduces new multiple recurrence results for disjoint systems and explores their implications for ergodic averages and Poincaré-type recurrence.

Findings

Established multiple recurrence for disjoint systems.

Derived Poincaré-type recurrence in metric spaces.

Proved uniform joint ergodicity when individual systems are ergodic.

Abstract

We consider mutually disjoint family of measure preserving transformations $T_1, \cdots, T_k$ on a probability space $(X, \mathcal{B}, \mu)$. We obtain the multiple recurrence property of $T_1, \cdots, T_k$ and this result is utilized to derive multiple recurrence of Poincar\'e type in metric spaces. We also present multiple recurrence property of Khintchine type. Further, we study multiple ergodic averages of disjoint systems and we show that $T_1, \cdots, T_k$ are uniformly jointly ergodic if each $T_i$ is ergodic.