Real almost reducibility of differentiable real quasi-periodic cocycles

TL;DR

This paper proves that smooth quasi-periodic cocycles with Diophantine frequencies can be approximated arbitrarily closely by reducible cocycles through real conjugations, up to a period doubling.

Contribution

It establishes the almost reducibility of infinitely differentiable quasi-periodic cocycles with real conjugations, extending previous results to the real setting.

Findings

Almost reducibility holds for smooth quasi-periodic cocycles under Diophantine conditions.

Real conjugations can be used to approximate the cocycles, up to period doubling.

The result applies to infinitely differentiable cocycles, broadening the scope of reducibility theory.

Abstract

We prove that infinitely differentiable almost reducible quasi-periodic cocycles, under a Diophantine condition on the frequency vector, are almost reducible to a sequence of real constant cocycles with a sequence of real conjugations, up to a period doubling.