Linearization of quasiperiodically forced circle flow beyond Brjuno condition

TL;DR

This paper proves that certain quasiperiodically forced circle flows are smoothly reducible to rotations when the base frequency is not super-Liouvillean and the fibered rotation number is Diophantine, extending previous results beyond the Brjuno condition.

Contribution

It establishes smooth reducibility for quasiperiodically forced circle flows beyond the Brjuno condition, showing local density of linearizable and mode-locking systems.

Findings

Systems with non-super-Liouvillean base frequency are smoothly reducible.

Linearizable and mode-locking systems are densely populated in the space of such flows.

The result extends reducibility beyond the classical Brjuno condition.

Abstract

We prove that an analytic quasiperiodically forced circle flow with a not super-Liouvillean base frequency and which is close enough to some constant rotation is $C^{\infty}$ rotations reducible, provided its fibered rotation number is Diophantine with respect to the base frequency. As a corollary, we obtain that among such systems, the linearizable ones and those displaying mode-locking are locally dense for the $C^\infty$-topology.