Non-conventional ergodicaverages and nilmanifolds

International audienceWe study the $L^2$-convergence of two types of ergodic averages.The first is the average of a product of functions evaluated at return timesalong arithmetic progressions, such as the expressions appearing inFurstenberg's proof of Szemer\'edi's Theorem. The second average is takenalong cubes whose sizes tend to$+\infty$.For each average, we show that it is sufficient to prove the convergence for special systems, the \emph{characteristic factors}.We build these factors in a general way, independent of the type of the average. To each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold. From the second convergence result we derive a combinatorial interpretationfor the arithmetic structure inside a set of integers ofpositive upper density