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 twofold 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 if and only if T̃ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if T̃ has zero topological entropy, and T has positive entropy if and only if 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 systems are also called chaotic in the sense of Devaney).