Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism

2025/09/10 by Gerber, Marlies, Kunde, Philipp
#03E15 #37A05 #37A25 #37C40 #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 37A35 #Secondary: 37A20

paper · doi:10.48550/arxiv.2509.09003

Abstract

We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a standard non-atomic probability space, is intractible. More precisely, the collection of isomorphic pairs of automorphisms in this class is not Borel, when considered as a subset of the Cartesian product of the collection of measure-preserving automorphisms with itself. This remains true if we restrict to zero-entropy mixing automorphisms that are also C^∞ diffeomorphisms of the five-dimensional torus. In addition, both of these results still hold if ``isomorphism'' is replaced by ``Kakutani equivalence.'' In our argument we show that for a uniquely and totally ergodic automorphism U and a particular family of automorphisms S, if T× U is isomorphic to T^-1× U with T\inS then T is isomorphic to T^-1. However, this type of ``cancellation'' of factors from isomorphic Cartesian products is not true in general. We present an example due to M. Lemańczyk of two weakly mixing automorphisms T and S and an irrational rotation R such that T× R is isomorphic to S× R, but T and S are not isomorphic.

Citations

• Classification of rank-one actions via the cutting-and-stacking parameters
• Non-Classifiability of Kolmogorov Diffeomorphisms up to Isomorphism
• Anti-classification results for weakly mixing diffeomorphisms
• Non-classifiability of Ergodic Flows up to Time Change
• Measure Preserving Diffeomorphisms of the Torus are Unclassifiable
• From Odometers to Circular Systems: A global structure theorem

Related

• Non-Classifiability of Kolmogorov Diffeomorphisms up to Isomorphism
• Anti-classification results for weakly mixing diffeomorphisms
• Facteurs disjoints des transformations melangeantes
• The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors
• Measure Preserving Diffeomorphisms of the Torus are Unclassifiable