A topological lens for a measure-preserving system

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We introduce a functor which associates to every measure preserving system (X,B,μ,T) a topological system (C₂(μ),T) 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 T. Some of our main results are as follows: (i) T is weakly mixing iff T is topologically transitive (iff it is topologically weakly mixing). (ii) T has zero entropy iff T has zero topological entropy, and T has positive entropy iff T 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 systems are also called chaotic in the sense of Devaney).