  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. The classes are canonically
isomorphic by a continuous map that takes factor maps to factor maps,
measure-isomorphisms to measure-isomorphisms, weakly mixing extensions to
weakly mixing extensions and compact extensions to compact extensions. The
first class includes all finite entropy ergodic transformations with an
odometer factor. By results in a previous paper, the second class contains all
transformations realizable as diffeomorphisms using the strongly uniform
untwisted Anosov-Katok method. An application of the main result will appear in
a forthcoming paper that shows that the diffeomorphisms of the torus are
inherently unclassifiable up to measure-isomorphism. Other consequences include
the existence measure distal diffeomorphisms of arbitrary countable distal
height.
