Article

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

02/2012;
Source: arXiv

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.

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##### Article:A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies
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ABSTRACT: We develop a new KAM scheme that applies to SL(2,R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's Theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters. Comment: 16 pages
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##### Chapter:Solving the Ten Martini Problem
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ABSTRACT: We discuss the recent proof of Cantor spectrum for the almost Mathieu operator for all conjectured values of the parameters.
09/2006: pages 5-16;
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##### Article:Almost localization and almost reducibility
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ABSTRACT: We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson's reducibility theory in this regime (our approach actually leads to improvements even in the perturbative regime: we are able to show, for all energies, ``almost reducibility'' in some band of analyticity). We also prove 1/2-H\"older continuity of the integrated density of states. For the almost Mathieu operator, our results hold through the entire regime of sub-critical coupling and imply also the dry version of the Ten Martini Problem for the concerned parameters.
06/2008;

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### Keywords

analytic quasi-periodic cocycle

analytic quasi-periodic linear system

Anderson localization results

equivalence

global reducibility results

Liouvillean frequency

local embedding theorem

new proof

quasi-periodic cocycles

quasi-periodic linear systems

quasi-periodic linear systems \cite{HoY}

recent

reducibility