Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

On Lyapunov exponents of continuous Schrödinger cocycles over irrational rotations

Authors: Wen Huang and Yingfei Yi
Journal: Proc. Amer. Math. Soc. 140 (2012), 1957-1962
MSC (2010): Primary 37B55; Secondary 37D25
Posted: September 26, 2011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider continuous, SL $(2,\mathbb{R})$ -valued, Schrödinger cocycles over irrational rotations. We prove two generic results on the Lyapunov exponents which improve the corresponding ones contained in a paper by Bjerklöv, Damanik and Johnson.

References

Bibliography

1. Artur Avila and Jairo Bochi, A uniform dichotomy for generic 𝑆𝐿(2,ℝ) cocycles over a minimal base, Bull. Soc. Math. France 135 (2007), no. 3, 407–417 (English, with English and French summaries). MR 2430187 (2009e:37013)
2. Kristian Bjerklöv, Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1015–1045. MR 2158395 (2007c:47035), http://dx.doi.org/10.1017/S0143385704000999
3. Kristian Bjerklöv, David Damanik, and Russell Johnson, Lyapunov exponents of continuous Schrödinger cocycles over irrational rotations, Ann. Mat. Pura Appl. (4) 187 (2008), no. 1, 1–6. MR 2346005 (2008j:37036), http://dx.doi.org/10.1007/s10231-006-0029-7
4. Jairo Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems 22 (2002), no. 6, 1667–1696. MR 1944399 (2003m:37035), http://dx.doi.org/10.1017/S0143385702001165
5. Jairo Bochi and Marcelo Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math. (2) 161 (2005), no. 3, 1423–1485. MR 2180404 (2006j:37060), http://dx.doi.org/10.4007/annals.2005.161.1423
6. J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152 (2000), no. 3, 835–879. MR 1815703 (2002h:39028), http://dx.doi.org/10.2307/2661356
7. David Damanik and Daniel Lenz, A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem, Duke Math. J. 133 (2006), no. 1, 95–123. MR 2219271 (2007a:37004), http://dx.doi.org/10.1215/S0012-7094-06-13314-8
8. David Damanik and Daniel Lenz, Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials, J. Math. Pures Appl. (9) 85 (2006), no. 5, 671–686 (English, with English and French summaries). MR 2229664 (2007a:47034), http://dx.doi.org/10.1016/j.matpur.2005.11.002
9. R. Fabbri, On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5 (2002), no. 1, 149–161 (English, with English and Italian summaries). MR 1881929 (2002m:34125)
10. R. Fabbri and R. Johnson, On the Lyapounov exponent of certain 𝑆𝐿(2,𝐑)-valued cocycles, Differential Equations Dynam. Systems 7 (1999), no. 3, 349–370. MR 1861078 (2002h:37041)
11. Michael Goldstein and Wilhelm Schlag, 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), no. 1, 155–203. MR 1847592 (2002h:82055), http://dx.doi.org/10.2307/3062114
12. Michael-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol′d et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58 (1983), no. 3, 453–502 (French). MR 727713 (85g:58057), http://dx.doi.org/10.1007/BF02564647
13. Russell A. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986), no. 1, 54–78. MR 818861 (87e:47065), http://dx.doi.org/10.1016/0022-0396(86)90125-7
14. Daniel Lenz, Uniform ergodic theorems on subshifts over a finite alphabet, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 245–255. MR 1889573 (2003j:37008), http://dx.doi.org/10.1017/S0143385702000111
15. Daniel Lenz, Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals, Comm. Math. Phys. 227 (2002), no. 1, 119–130. MR 1903841 (2003k:47043), http://dx.doi.org/10.1007/s002200200624
16. Eugene Sorets and Thomas Spencer, Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials, Comm. Math. Phys. 142 (1991), no. 3, 543–566. MR 1138050 (93b:81058)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37B55, 37D25

Retrieve articles in all journals with MSC (2010): 37B55, 37D25

Additional Information

Wen Huang
Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei Anhui 230026, People’s Republic of China
Email: wenh@mail.ustc.edu.cn

Yingfei Yi
Affiliation: School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China – and – School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: yi@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11042-6
PII: S 0002-9939(2011)11042-6
Keywords: Lyapunov exponent, Schrödinger cocycles, non-uniform hyperbolicity
Received by editor(s): December 5, 2010
Received by editor(s) in revised form: January 30, 2011
Posted: September 26, 2011
Additional Notes: The first author is partially supported by NSFC(10911120388,11071231), Fok Ying Tung Education Foundation and the Fundamental Research Funds for the Central Universities (WK0010000001,WK0010000014).
The second author is partially supported by NSF grant DMS0708331, NSFC Grant 10428101, and a Changjiang Scholarship from Jilin University
Communicated by: Bryna Kra
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|