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Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles

Published online by Cambridge University Press:  30 September 2009

S. JITOMIRSKAYA ,
D. A. KOSLOVER  and
M. S. SCHULTEIS
S. JITOMIRSKAYA
Affiliation:
Department of mathematics, University of California, Irvine, CA 92697, USA (email: [email protected])
D. A. KOSLOVER
Affiliation:
Department of mathematics, University of California, Irvine, CA 92697, USA (email: [email protected]) Department of mathematics, University of Texas at Tyler, Tyler, TX 75799, USA (email: [email protected])
M. S. SCHULTEIS
Affiliation:
Department of mathematics, University of California, Irvine, CA 92697, USA (email: [email protected]) Department of mathematics, Concordia University, Irvine, CA 92612, USA (email: [email protected])
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Abstract

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1881 - 1905
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Avila, A. and Jitomirskaya, S.. The ten Martini problem. Ann. of Math. (2) 170(1) (2009), 303–342.CrossRefGoogle Scholar
[2]Bochi, J.. Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles. Preprint.Google Scholar
[3]Bochi, J.. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 22 (2002), 1667–1696.CrossRefGoogle Scholar
[4]Bochi, J. and Viana, M.. The Lyapunov exponents of generic volume preserving and symplectic maps. Ann. of Math. (2) 161(3) (2005), 1423–1485.CrossRefGoogle Scholar
[5]Bourgain, J.. Green’s Function Estimates for Lattice Schrödinger Operators and Applications (Annals of Mathematics Studies, 158). Princeton University Press, Princeton, NJ, 2005, p. 173.CrossRefGoogle Scholar
[6]Bourgain, J.. Positivity and continuity of the Lyapunov exponent for shifts on 𝕋d with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96 (2005), 313–355.CrossRefGoogle Scholar
[7]Bourgain, J. and Goldstein, M.. On nonperturbative localization with quasi-periodic potential. Ann. of Math. (2) 152 (2000), 835–879.CrossRefGoogle Scholar
[8]Bourgain, J., Goldstein, M. and Schlag, W.. Anderson localization for Schrödinger operators on ℤ with potentials given by the skew-shift. Comm. Math. Phys. 220(3) (2001), 583–621.CrossRefGoogle Scholar
[9]Bourgain, J. and Jitomirskaya, S.. Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Stat. Phys. 108(5/6) (2002), 1028–1218.CrossRefGoogle Scholar
[10]Damanik, D. and Lenz, D.. A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem. Duke Math. J. 133(1) (2006), 95–123.CrossRefGoogle Scholar
[11]Furman, A.. On the multiplicative ergodic theorem for uniquely ergodic systems. Ann. Inst. H. Poincaré 33 (1997), 797–815.CrossRefGoogle Scholar
[12]Germinet, F. and Jitomirskaya, S.. Strong dynamical localization for the almost Mathieu model. Rev. Math. Phys. 13 (2001), 755–765.CrossRefGoogle Scholar
[13]Goldshtein, M. and Schlag, W.. Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2) 154 (2001), 155–203.CrossRefGoogle Scholar
[14]Jitomirskaya, S.. Metal-insulator transition for the almost Mathieu operator. Ann. of Math. (2) 150 (1999), 1159–1175.CrossRefGoogle Scholar
[15]Jitomirskaya, S., Koslover, D. A. and Schulteis, M.. Localization for a family of one-dimensional quasiperiodic operators of magnetic origin. Ann. Inst. H. Poincaré 6 (2005), 103–121.CrossRefGoogle Scholar
[16]Klein, S.. Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function. J. Funct. Anal. 218 (2005), 255–292.CrossRefGoogle Scholar
[17]Knill, O.. The upper Lyapunov exponent of SL(2,ℝ) cocycles: discontinuity and the problem of positivity. Lyapunov Exponents (Oberwolfach, 1990) (Lecture Notes in Mathematics, 1486). Springer, Berlin, 1991, pp. 86–97.CrossRefGoogle Scholar
[18]Last, Y.. Spectral theory of Sturm–Louiville operators on infinite intervals: a review of recent developments. Sturm–Louiville Theory. Birkhäuser, Basel, 2005, pp. 99–120.CrossRefGoogle Scholar
[19]Lojasiewicz, S.. Sur le probléme de la division. Studia Math. 18 (1959), 87–136.CrossRefGoogle Scholar
[20]Simon, B.. Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory (AMS Colloquium Publications). American Mathematical Society, Providence, RI, 2005.Google Scholar
[21]Thouless, D.. Bandwidth for a quasiperiodic tight-binding model. Phys. Rev. 28 (1983), 4272–4276.CrossRefGoogle Scholar
[22]Thouvenot, J.-P.. An example of discontinuity in the computation of the Lyapunov exponents. Proc. Steklov Inst. Math. 216 (1997), 366–369.Google Scholar
[23]Young, L.-S.. Some open sets of non-uniformly hyperbolic cocycles. Ergod. Th. & Dynam. Sys. 13 (1993), 409–415.CrossRefGoogle Scholar