The Rotations Reductibility of Quasi-periodic Linear Systems with Weak-Liouvillean Frequency

Abstract

This paper studies the rotations reducibility of a class of quasi-periodic linear sl(2, ℝ) systems with a usual frequency ω ∈ ℝ^1+d. In fact we proved that if the frequency satisfies some weak-Liouvillean conditions, the system is positive-measure rotations reducible. We will prove this result via a generalized KAM iteration.

Keywords

KAM theory / rotations reducibility / weak-Liouvillean frequency / 37C55

References

[1] Avila A, Fayad B, Krikorian R. A KAM scheme for SL(2,ℝ) cocycles with Liouvillean frequencies. Geom. Funct. Anal., 2011, 21(5): 1001-1019

[2] Dinaburg EI, Sinaĭ JaG. The one-dimensional Schrödinger equation with quasiperiodic potential. Funkcional. Anal. i Priložen, 1975, 9(4): 8-21

[3] Eliasson LH. Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Comm. Math. Phys., 1992, 146(3): 447-482

[4] Ge L. On the almost reducibility conjecture. Geom. Funct. Anal., 2024, 34(1): 32-59

[5] Ge L, You J. Arithmetic version of Anderson localization via reducibility. Geom. Funct. Anal., 2020, 30(5): 1370-1401

[6] Hou X., Wang J., Zhou Q., Absolutely continuous spectrum of multifrequency quasiperiodic Schrödinger operator. J. Funct. Anal., 2020, 279(6): Paper No. 108632, 33 pp.

[7] Hou X, You J. Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math., 2012, 190(1): 209-260

[8] Krikorian R., Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts. Astérisque, 1999, 259: vi+216 pp.

[9] Krikorian R. Réductibilité presque partout des flots fibres quasi-périodiques à valeurs dans des groupes compacts. Ann. Sci. École Norm. Sup. (4), 1999, 32(2): 187-240

[10] Krikorian R. Global density of reducible quasi-periodic cocycles on T¹SU(2). Ann. of Math. (2), 2001, 154(2): 269-326

[11] Krikorian R, Wang J, You J, Zhou Q. Linearization of quasiperiodically forced circle flows beyond Brjuno condition. Comm. Math. Phys., 2018, 358(1): 81-100

[12] Liang Z, Zhao Z, Zhou Q. 1-d quantum harmonic oscillator with time quasi-periodic quadratic perturbation: reducibility and growth of Sobolev norms. J. Math. Pures Appl. (9), 2021, 146: 158-182

[13] Liang Z., Zhao Z., Zhou Q., Almost reducibility and oscillatory growth of Sobolev norms. Adv. Math., 2024, 436: Paper No. 109417, 54 pp.

[14] Liu J, Shan Y, Wang J. Reducibility of three dimensional skew symmetric system with high dimensional weak Liouvillean frequencies. Acta Math. Sin. (Engl. Ser.), 2024, 40(10): 2388-2410

[15] Marx CA, Jitomirskaya S. Dynamics and spectral theory of quasi-periodic Schrödinger-type operators. Ergodic Theory Dynam. Systems, 2017, 37(8): 2353-2393

[16] Pöschel J., A lecture on the classical KAM theorem. 2009, arXiv:0908.2234

[17] Rüssmann H. On the one-dimensional Schrödinger equation with a quasiperiodic potential. Nonlinear Dynamics, 1980, New York, New York Acad. Sci.90107357

[18] Wang J, You J. Boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequency. J. Differential Equations, 2016, 261(2): 1068-1098

[19] Wang J, You J, Zhou Q. Response solutions for quasi-periodically forced harmonic oscillators. Trans. Amer. Math. Soc., 2017, 369(6): 4251-4274

[20] You J, Zhou Q. Phase transition and semi-global reducibility. Comm. Math. Phys., 2014, 330(3): 1095-1113

[21] Zhang D, Wu H. On the reducibility of two-dimensional quasi-periodic systems with Liouvillean basic frequencies and without non-degeneracy condition. J. Differential Equations, 2022, 324: 1-40

[22] Zhou Q, Wang J. Reducibility results for quasiperiodic cocycles with Liouvillean frequency. J. Dynam. Differential Equations, 2012, 24(1): 61-83

RIGHTS & PERMISSIONS

Peking University