2/22/2006

Bernard Host
Universit´e de Marne la Vall´ee

Multiple correlations in Ergodic Theory

Consider a probability space (X,\CB,\mu) endowed with an invertible measure preserving transformation T. This system is "mixing'" if for any sets A,B\in \CB the measure of A\cap T^nB converges to the product \mu(A)\mu(B) when n goes to infinity. It is called `"three fold mixing'" if for any sets A,B,C\in \CB we have \mu(A\cap T^mB \cap T^n C)\to\mu(A)\mu(B)\mu(C)when m and n-m\to+\infty.
Does mixing imply three fold mixing? This question is almost as old as ergodic theory itself and still open. Very little progress has been done  until now. I will first present some relatively old results.
This question is related to the notion of `"multicorrelations." If f is a function on X, its `"correlation sequence'' c_n(f):=\int f(x) f(T^nx) d\mu(x) has a nice description as the Fourier transform of a measure. But higher correlation sequences, like c_{m,n}(f):=\int  f(x) f(T^mx)f(T^nx) dx, are much less understood. I plan to present a few properties of these sequences.