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

By Jiangong You, Qi Zhou. 2012

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In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincaré map of an analytic quasi-periodic linear system close to constant, which bridges both methods and results in quasi-periodic linear systems and cocycles. We also show that the almost reducibility of an analytic quasi-periodic linear system is equivalent to the almost reducibility of its corresponding Poincaré cocycle. By the local embedding theorem and the equivalence, we transfer the recent local almost reducibility results of quasi-periodic linear systems (Hou and You, in Invent Math 190:209–260, 2012) to quasi-periodic cocycles, and the global reducibility results of quasi-periodic cocycles (Avila, in Almost reducibility and absolute continuity, 2010; Avila et al., in Geom Funct Anal 21:1001–1019, 2011) to quasi-periodic linear systems. Finally, we give a positive answer to a question of Avila et al. (Geom Funct Anal 21:1001–1019, 2011) and use it to study point spectrum of long-range quasi-periodic operator with Liouvillean frequency. The embedding also holds for some nonlinear systems.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/39d31459c02290613378ed2fb496499c619468d2

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In Geometry and Topology

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Citations in the corpus

Sharp phase transitions for the almost Mathieu operator (2015, background): “Conversely, to deal with localization problem by reducibility was first realized by You-Zhou in [40].”