Rank-one systems, flexible classes and Shannon orbit equivalence

TL;DR

This paper establishes Shannon orbit equivalences between the universal odometer and various rank-one systems using flexible classes, revealing that properties like mixing or eigenvalues are not preserved under such equivalences.

Contribution

It introduces flexible classes of rank-one transformations and proves they contain elements Shannon orbit equivalent to the universal odometer, generalizing previous results.

Findings

Flexible classes include odometers, Chacon's map, and systems with specific eigenvalues.

Strong mixing and eigenvalue properties are not preserved under Shannon orbit equivalence.

The main construction provides explicit transformations in certain cases.

Abstract

We build a Shannon orbit equivalence between the universal odometer and a variety of rank-one systems. This is done in a unified manner, using what we call flexible classes of rank-one transformations. Our main result is that every flexible class contains an element which is Shannon orbit equivalent to the universal odometer. Since a typical example of flexible class is $\{T\}$ when $T$ is an odometer, our work generalizes a recent result by Kerr and Li, stating that every odometer is Shannon orbit equivalent to the universal odometer. When the flexible class is a singleton, the rank-one transformation given by the main result is explicit. This applies to odometers and Chacon's map. We also prove that strongly mixing systems, systems with a given eigenvalue, or irrational rotations whose angle belongs to any fixed open subset of the real line form flexible classes. In particular, strong mixing, rationality or irrationality of the eigenvalues are not preserved under Shannon orbit equivalence.