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The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings that preserve underlying timing factors. The classes are canonically isomorphic by a continuous map that takes synchronous and anti-synchronous factor maps to synchronous and anti-synchronous factor maps, synchronous and anti-synchronous measure-isomorphisms to synchronous and anti-synchronous measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. The first class includes all finite entropy ergodic transformations that have an odometer factor. By results in [ 6 ], the second class contains all transformations realizable as diffeomorphisms using the untwisted Anosov–Katok method. An application of the main result will appear in a forthcoming paper [ 7 ] that shows that the diffeomorphisms of the torus are inherently unclassifiable up to measure-isomorphism. Other consequences include the existence of measure distal diffeomorphisms of arbitrary countable distal height.

From odometers to circular systems: A global structure theorem