Reducibility of ultra-differentiable quasiperiodic cocycles under an adapted arithmetic condition

Bounemoura, Abed; Chavaudret, Claire; Liang, Shuqing

doi:10.1090/proc/15433

Reducibility of ultra-differentiable quasiperiodic cocycles under an adapted arithmetic condition
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by Abed Bounemoura, Claire Chavaudret and Shuqing Liang

Proc. Amer. Math. Soc. 149 (2021), 2999-3012

DOI: https://doi.org/10.1090/proc/15433

Published electronically: April 29, 2021

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Abstract:

We prove a reducibility result for $sl(2,\mathbb {R})$ quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-Rüssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition.

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