Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications

TL;DR

This paper establishes a local embedding theorem linking analytic quasi-periodic cocycles and linear systems, enabling transfer of reducibility results and proving Anderson localization for certain operators.

Contribution

The paper introduces a new local embedding theorem that connects quasi-periodic cocycles with linear systems, facilitating the transfer of reducibility results and solving an open question.

Findings

Equivalence between almost reducibility of cocycles and linear systems.

Transfer of local and global reducibility results.

Proof of Anderson localization for long-range operators with Liouvillean frequency.

Abstract

In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincar\'{e} map of an analytic quasi-periodic linear system close to constant. With this local embedding theorem, we get fruitful new results. We show that the almost reducibility of an analytic quasi-periodic linear system is equivalent to the almost reducibility of its corresponding Poincar\'e cocycle. By the local embedding theorem and the equivalence, we transfer the recent local almost reducibility results of quasi-periodic linear systems \cite{HoY} to quasi-periodic cocycles, and the global reducibility results of quasi-periodic cocycles \cite{A,AFK} to quasi-periodic linear systems. Finally, we give a positive answer to a question of \cite{AFK} and use it to prove Anderson localization results for long-range quasi-periodic operator with Liouvillean frequency, which gives a new proof of \cite{AJ05,AJ08,BJ02}. The method developed in our paper can also be used to prove some nonlinear local embedding results.