From odometers to circular systems: A global structure theorem

Matthew Foreman, ,Department of Mathematics, University of California, Irvine, CA 92697, USA, Benjamin Weiss, ,Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
2019 Journal of Modern Dynamics  
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
more » ... eisomorphisms, 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 measureisomorphism. Other consequences include the existence of measure distal diffeomorphisms of arbitrary countable distal height. CONTENTS
doi:10.3934/jmd.2019024 fatcat:gk4hebl2qbhxzc7khsufx7i2jm