# From Odometers to Circular Systems: A global structure theorem

## TL;DR

This paper establishes a structural equivalence between two large classes of ergodic measure-preserving systems, revealing deep connections and implications for classifying complex dynamical systems like toral diffeomorphisms.

## Contribution

It proves a canonical isomorphism between Odometer Based and Circular Systems, linking their structures and extensions, with implications for classifying diffeomorphisms.

## Key findings

- Both classes have the same global structure with respect to joinings.
- The classes are canonically isomorphic via a continuous map.
- Implications include the unclassifiability of torus diffeomorphisms up to measure-isomorphism.

## Abstract

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.