Abstract
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.
Similar content being viewed by others
References
Aubry, S., André, G.: Analyticity breaking and Anderson localization in incommensurate lattices. In: Group Theoretical Methods in Physics (Proc. Eighth Internat. Colloq. Kiryat Anavim, 1979), Bristol: Hilger, 1980, pp. 133–164
Avila A.: Density of positive Lyapunov exponents for quasiperiodic \({SL(2, \mathbb{R})}\) cocycles in arbitrary dimension. J. Mod. Dyn. 3, 629–634 (2009)
Avila, A.: Global theory of one-frequency Schrödinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity. Preprint, http://arxiv.org/abs/0905.3902v1 [math.DS], 2009
Avila, A.: Almost reducibility and absolute continuity. Preprint, http://arxiv.org/abs/1006.0704v1 [math.DS], 2010
Avila A., Fayad B., Krikorian R.: A KAM scheme for \({{\rm SL}(2,\mathbb{R})}\) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21, 1001–1019 (2011)
Avila A., Jitomirskaya S.: The Ten Martini Problem. Ann. Math. 170, 303–342 (2009)
Avila A., Jitomirskaya S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)
Avila A., Krikorian R.: educibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)
Berti M., Biasco L.: Forced vibrations of wave equations with non-monotone nonlinearities. Ann. I. H. Poincaré-AN. 23, 439–474 (2006)
Bourgain J.: On the spectrum of lattice Schrodinger operators with deterministic potential II. Dedicated to the memory of Tom Wolff. J. Anal. Math. 88, 221–254 (2002)
Bourgain J.: Positivity and continuity of the Lyapunov exponent for shifts on \({\mathbb{T}^d}\) with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96, 313–355 (2005)
Bourgain J., Goldstein M.: On nonperturbative localization with quasiperiodic potential. Annals of Math. 152, 835–879 (2000)
Bourgain J., Jitomirskaya S.: Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148(3), 453–463 (2002)
Chavaudret C.: Reducibility of quasiperiodic cocycles in linear Lie groups. Erg. Th. Dyn. Sys. 31(03), 741–769 (2011)
Chavaudret, C.: Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles. http://arxiv.org/abs/0912.4814v3 [math.DS], 2010 to appear in Bulletin de la Société Mathématique de France
Deimling, K.: Nonlinear functional analysis. Berlin-Heidelberg-NewYork: Springer-Verlag, 1985
Dias J.L.: A normal form theorem for Brjuno skew systems through renormalization. J. Diff. Eq. 230, 1–23 (2006)
Dinaburg E., Sinai Ya.: The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9, 279–289 (1975)
Eliasson H.: Floquet solutions for the one-dimensional quasiperiodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)
Fayad, B., Katok, A., Windsor, A.: Mixed spectrum reparameterizations of linear flows on \({\mathbb{T}^2}\) . Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1(4), 521–537. (2001)
Fayad B., Krikorian R.: Rigidity results for quasiperiodic \({{\rm SL}(2,\mathbb{R})}\) -cocycles. J. Mod. Dyn. 3(4), 479–510 (2009)
Gordon A.Y.: The point spectrum of one-dimensional Schrödinger operator (Russian). Usp. Mat. Nauk. 31, 257–258 (1976)
Gordon A.Y., Jitomirskaya S., Last Y., Simon B.: Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178, 169–183 (1997)
Hou X., You J.: Almost reducibility and non-perturbative reducibility of quasiperiodic linear systems. Invent. Math. 190, 209–260 (2012)
Jitomirskaya S.: Metal-Insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159–1175 (1999)
Johnson R.: Analyticity of spectral subbundles. J. Diff. Eqs. 35(3), 366–387 (1980)
Johnson R., Sell G.: Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems. J. Diff. Eqs. 41(2), 262–288 (1981)
Johnson R., Moser J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84(3), 403–438 (1982)
Kotani, S.: Lyaponov indices determine absolutely continuous spectra of stationary random onedimensional Schrondinger operators. In: Stochastic Analysis, K. Ito, ed., Amsterdam: North Holland, 1984, pp. 225–248
Krikorian R.: Global density of reducible quasi-periodic cocycles on \({\mathbb{T}^1\times SU(2)}\) . Ann. Math. 2, 269–326 (2001)
Krikorian, R.: Reducibility, differentiable rigidity and Lyapunov exponents for quasiperiodic cocycles on \({\mathbb{T} \times SL(2,\mathbb{R})}\). Preprint, http://arxiv.org/abs/math/0402333v1 [math.DS], 2004
Krikorian, R., Wang, J., You, J., Zhou, Q.: Linearization of quasiperiodically forced circle flow beyond Brjuno condition. Preprint
Kuksin, S., Pöschel, J.: On the Inclusion of Analytic Symplectic Maps in Analytic Hamiltonian Flows and Its Applications. In: Seminar on Dynamical Systems, S. Kuksin, V. Lazutkin, J. Pöchel eds., Basel: Birkhäuser, 1994, pp. 96–116
Maslov V.P., Molchanov S.A., Gordon A.Y.: Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture. Russ. J. Math. Phys. 1, 71–104 (1993)
Moser J., Poschel J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helv. 59, 39–85 (1984)
Puig J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19(2), 355–376 (2006)
Rychlik M.: Renormalization of cocycles and linear ODE with almost-periodic coefficients. Invent. Math. 110, 173–206 (1992)
Simon B.: Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
Rights and permissions
About this article
Cite this article
You, J., Zhou, Q. Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications. Commun. Math. Phys. 323, 975–1005 (2013). https://doi.org/10.1007/s00220-013-1800-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-013-1800-4