A topological lens for a measure-preserving system

We introduce a functor which associates to every measure preserving system (X,B, μ, T ) a topological system (C2(μ), T̃ ) defined on the space of 2-fold couplings of μ, called the topological lens of T . We show that often the topological lens “magnifies” the basic measure dynamical properties of T in terms of the corresponding topological properties of T̃ . Some of our main results are as follows: (i) T is weakly mixing iff T̃ is topologically transitive (iff it is topologically weakly mixing). (ii) T has zero entropy iff T̃ has zero topological entropy, and T has positive entropy iff T̃ has infinite topological entropy. (iii) For T a K-system, the topological lens is a P -system (i.e. it is topologically transitive and the set of periodic points is dense; such sytems are also called chaotic in the sense of Devaney). 2000 Mathematical Subject Classification: 37A05, 37A35, 37B05, 37B40