Note sur la continuit\'e de la projection dans les facteurs de Host-Kra

TL;DR

This paper investigates the relationship between the continuity of projections in characteristic factors and the behavior of cubic ergodic averages, providing insights into multiple recurrence problems in ergodic theory.

Contribution

It demonstrates that the continuity of the projection in characteristic factors characterizes cubic averages, linking structural properties to ergodic averages.

Findings

Continuity of projections characterizes cubic averages.

Provides a new perspective on multiple recurrence problems.

Connects structural properties of systems to ergodic behavior.

Abstract

B. Host and B. Kra (2005) have introduced the characteristic factors for studying cubic ergodic averages. These factors allow one to resolve, in particular, multiple recurrence problems introduced by H. Furstenberg (1977). We show here that the continuity of the projection of the system in its characteristic factors characterises cubic averages.