Cocycle conjugacy classes of binary shifts

TL;DR

The paper demonstrates that all binary shifts on the hyperfinite II_1 factor are cocycle conjugate to infinitely many non-conjugate shifts, revealing a rich structure of these automorphisms.

Contribution

It establishes the existence of infinitely many non-conjugate binary shifts cocycle conjugate to each other on the hyperfinite II_1 factor, especially for those with infinite commutant index.

Findings

Every binary shift is cocycle conjugate to countably many non-conjugate shifts.

This applies notably to shifts with infinite commutant index.

The result uncovers a complex classification landscape for binary shifts.

Abstract

We show that every binary shift on the hyperfinite $II_1$ factor $R$ is cocycle conjugate to at least countably many non-conjugate binary shifts. This holds in particular for binary shifts of infinite commutant index.