Multiple mixing in ergodic theory

Abstract

In 1949 V.A. Rokhlin introduced into ergodic theory the k-fold mixing and puzzled the mathematical community with the problem of the mismatch of these invariants. Here's what Rokhlin wrote: "The proposed work arose from the author's attempts to solve the well-known spectral problem of the theory of dynamical systems: are there any metrically different dynamical systems with the same continuous (in particular, Lebesgue) spectrum? Using new metric invariants introduced for this purpose, the author tried to find among the ergodic automorphisms of compact commutative groups automorphisms of various metric types. It turned out, however, that for all the indicated automorphisms the new invariants are exactly the same." We discuss this long time open problem.