A circle diffeomorphism with breaks that is smoothly linearizable

TL;DR

This paper constructs a specific example of a circle diffeomorphism with breaks, demonstrating it can be smoothly conjugate to a rotation even with breaks on distinct trajectories, and the invariant measure is absolutely continuous.

Contribution

It provides the first explicit example of a smooth conjugation for a circle diffeomorphism with breaks on distinct trajectories, expanding understanding of linearizability conditions.

Findings

Constructed a piecewise linear circle homeomorphism with four break points on distinct trajectories.

Proved the invariant measure is absolutely continuous with respect to Lebesgue measure.

Demonstrated the rotation number can be a Roth number not of bounded type.

Abstract

In this paper we answer positively a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case when its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories, and whose invariant measure is absolutely continuous w.r.t.\ the Lebesgue measure. The irrational rotation number for our example can be chosen a Roth number, but not of bounded type.