Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups

TL;DR

This paper proves that quasiperiodic cocycles close to constants in compact Lie groups exhibit differentiable rigidity, meaning measurable conjugacy implies smooth conjugacy under Diophantine conditions, with the K.A.M. scheme providing explicit conjugations.

Contribution

It establishes differentiable rigidity results for quasiperiodic cocycles in compact Lie groups, extending previous work by linking measurable and smooth conjugacies via K.A.M. schemes.

Findings

Measurable conjugacy implies $C^{ abla}$-conjugacy for certain cocycles.

The K.A.M. scheme produces explicit smooth conjugations.

Global rigidity holds under renormalization convergence.

Abstract

We study close-to-constants quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $d \in \mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{\infty}$-conjugate to it, and the K.A.M. scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.