Furstenberg counterexamples over Diophantine rotations

TL;DR

This paper constructs specific cocycles over Diophantine rotations that are minimal but not uniquely ergodic, demonstrating their density in certain open sets and under full-measure conditions, thus providing new counterexamples in dynamical systems.

Contribution

It introduces a method to construct dense sets of minimal, non-uniquely ergodic cocycles over Diophantine rotations, expanding understanding of ergodic properties in these systems.

Findings

Cocycles are dense in an open subset of the cocycle space.

Such cocycles are dense under full-measure arithmetic conditions.

Constructed cocycles are minimal but not uniquely ergodic.

Abstract

We construct cocycles in $\mathbb{T} \times SU(2)$ over Diophantine rotations that are minimal and not uniquely ergodic. Such cocycles are dense in an open subset of cocycles over the fixed Diophantine rotation. By a standard argument, they are dense in the whole set of such cocycles if the rotation satisfies a full-measure arithmetic condition.