Furstenberg's famous proof of Szemeredi's theorem leads to a natural
question about the convergence and limit of some multiple ergodic
averages. In the case of Z-actions these averages were studied by
Host-Kra and Ziegler. They show that the limiting behavior of such
multiple ergodic average is determined on a certain factor that can be
given the structure of an inverse limit of nilsystems (i.e. rotations
on a nilmanifold). This structure result can be generalized to Z^d
actions (where the average is taken over a Folner sequence), but the
non-finitely generated case is still open (at least from
ergodic-theoretical point of view). The only progress prior to our
work is due to Bergelson Tao and Ziegler, who studied actions of the
infinite direct sum of Z/pZ. In our work we generalize this further to
the case where the sum is taken over different primes (the most
interesting case is when the multiset of primes is unbounded). We will
explain how this case is significantly different from the work of
Bergelson Tao and Ziegler by describing a new phenomenon that only
happens in these settings. Moreover, we will discuss a generalized
version of nilsystems that plays a role in our work and some
corollaries. If time allows we will also discuss the group actions of
other abelian groups.
Meeting ID: 838 4895 5300
Passcode: 797123