Monotonic cocycles

• 成果类型: Article
• 署名作者: Avila, Artur; Krikorian, Raphael
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Instituto Nacional de Matematica Pura e Aplicada (IMPA); Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0572-6
• 发表日期: 2015
• 页码: 271-331
• 关键词: absolutely continuous-spectrum; quasi-periodic operators; lyapunov exponents; jacobi matrices; linear-systems; reducibility; continuity; EQUATIONS; DYNAMICS; THEOREM

摘要

We develop a local theory of multidimensional quasiperiodic cocycles which are not homotopic to a constant. It describes a -open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on the frequency, being much more robust than the local theory of cocycles homotopic to a constant. Our analysis is centered around the notion of monotonicity with respect to some dynamical variable. For such monotonic cocycles, we obtain a sharp rigidity result, minimality of the projective action, typical nonuniform hyperbolicity, and a surprising result of smoothness of the Lyapunov exponent (while no better than Holder can be obtained in the case of cocycles homotopic to a constant, and only under arithmetic restrictions). Our work is based on complexification ideas, extended A la Lyubich to the smooth setting (through the use of asymptotically holomorphic extensions). We also develop a counterpart of this theory centered around the notion of monotonicity with respect to a parameter variable, which applies to the analysis of cocycles over more general dynamical systems and generalizes key aspects of Kotani Theory. We conclude with a more detailed discussion of one-dimensional monotonic cocycles, for which results about rigidity and typical nonuniform hyperbolicity can be globalized using a new result about convergence of renormalization.