Density of positive Lyapunov exponents for $\mathrm {SL}(2,\mathbb {R})$-cocycles
HTML articles powered by AMS MathViewer
- by Artur Avila;
- J. Amer. Math. Soc. 24 (2011), 999-1014
- DOI: https://doi.org/10.1090/S0894-0347-2011-00702-9
- Published electronically: April 8, 2011
- PDF | Request permission
Abstract:
We show that $\mathrm {SL}(2,\mathbb {R})$-cocycles with a positive Lyapunov exponent are dense in all regularity classes and for all non-periodic dynamical systems. For Schrödinger cocycles, we show prevalence of potentials for which the Lyapunov exponent is positive for a dense set of energies.References
- Artur Avila, Density of positive Lyapunov exponents for quasiperiodic $\textrm {SL}(2,\Bbb R)$-cocycles in arbitrary dimension, J. Mod. Dyn. 3 (2009), no. 4, 631â636. MR 2587090, DOI 10.3934/jmd.2009.3.631
- Artur Avila, On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators, Comm. Math. Phys. 288 (2009), no. 3, 907â918. MR 2504859, DOI 10.1007/s00220-008-0667-2
- Avila, A., On the Kotani-Last and the Schrödinger Conjectures. In preparation.
- Artur Avila and Jairo Bochi, A formula with some applications to the theory of Lyapunov exponents, Israel J. Math. 131 (2002), 125â137. MR 1942304, DOI 10.1007/BF02785853
- Artur Avila and David Damanik, Generic singular spectrum for ergodic Schrödinger operators, Duke Math. J. 130 (2005), no. 2, 393â400. MR 2181094, DOI 10.1215/S0012-7094-05-13035-6
- Avila, A.; Krikorian, R., Monotonic cocycles. In preparation.
- J. Avron, P. H. M. van Mouche, and B. Simon, On the measure of the spectrum for the almost Mathieu operator, Comm. Math. Phys. 132 (1990), no. 1, 103â118. MR 1069202, DOI 10.1007/BF02278001
- Joseph Avron and Barry Simon, Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J. 50 (1983), no. 1, 369â391. MR 700145, DOI 10.1215/S0012-7094-83-05016-0
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
- Jairo Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems 22 (2002), no. 6, 1667â1696. MR 1944399, DOI 10.1017/S0143385702001165
- Jairo Bochi and Marcelo Viana, Lyapunov exponents: how frequently are dynamical systems hyperbolic?, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 271â297. MR 2090775
- Christian Bonatti, Xavier GĂłmez-Mont, and Marcelo Viana, GĂ©nĂ©ricitĂ© dâexposants de Lyapunov non-nuls pour des produits dĂ©terministes de matrices, Ann. Inst. H. PoincarĂ© C Anal. Non LinĂ©aire 20 (2003), no. 4, 579â624 (French, with English and French summaries). MR 1981401, DOI 10.1016/S0294-1449(02)00019-7
- David Damanik, Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simonâs 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 539â563. MR 2307747, DOI 10.1090/pspum/076.2/2307747
- E. I. Dinaburg and Ja. G. SinaÄ, The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. i PriloĆŸen. 9 (1975), no. 4, 8â21 (Russian). MR 470318
- Bassam Fayad and RaphaĂ«l Krikorian, Rigidity results for quasiperiodic $\textrm {SL}(2,\Bbb R)$-cocycles, J. Mod. Dyn. 3 (2009), no. 4, 497â510. MR 2587083, DOI 10.3934/jmd.2009.3.479
- Brian R. Hunt, Tim Sauer, and James A. Yorke, Prevalence: a translation-invariant âalmost everyâ on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217â238. MR 1161274, DOI 10.1090/S0273-0979-1992-00328-2
- Svetlana Jitomirskaya, Ergodic Schrödinger operators (on one foot), Spectral theory and mathematical physics: a Festschrift in honor of Barry Simonâs 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 613â647. MR 2307750, DOI 10.1090/pspum/076.2/2307750
- A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2) 110 (1979), no. 3, 529â547. MR 554383, DOI 10.2307/1971237
- A. N. Kolmogorov, ThĂ©orie gĂ©nĂ©rale des systĂšmes dynamiques et mĂ©canique classique, Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 1, Erven P. Noordhoff N. V., Groningen, 1957, pp. 315â333 (French). MR 97598
- Shinichi Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 225â247. MR 780760, DOI 10.1016/S0924-6509(08)70395-7
- Jean-Christophe Yoccoz, Some questions and remarks about $\textrm {SL}(2,\mathbf R)$ cocycles, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 447â458. MR 2093316
- Marcelo Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math. (2) 167 (2008), no. 2, 643â680. MR 2415384, DOI 10.4007/annals.2008.167.643
Bibliographic Information
- Artur Avila
- Affiliation: Institut de Mathématiques de Jussieu, CNRS UMR 7586, 175 rue du Chevaleret, 75013, Paris, France; IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
- Email: artur@math.sunysb.edu
- Received by editor(s): May 25, 2010
- Received by editor(s) in revised form: August 2, 2010, and March 22, 2011
- Published electronically: April 8, 2011
- Additional Notes: This research was partially conducted during the period when the author was a Clay Research Fellow
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 24 (2011), 999-1014
- MSC (2010): Primary 37H15
- DOI: https://doi.org/10.1090/S0894-0347-2011-00702-9
- MathSciNet review: 2813336