Exponential dichotomy of linear cocycles over irrational rotations

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Exponential dichotomy of linear cocycles over irrational rotations. / Ivanov, Alexey V.

Proceedings of the International Conference Days on Diffraction 2020, DD 2020. ed. / O.V. Motygin; A.P. Kiselev; L.I. Goray; T.M. Zaboronkova; A.Ya. Kazakov; A.S. Kirpichnikova. Institute of Electrical and Electronics Engineers Inc., 2020. p. 38-43 9274638 (Proceedings of the International Conference Days on Diffraction 2020, DD 2020).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Harvard

Ivanov, AV 2020, Exponential dichotomy of linear cocycles over irrational rotations. in OV Motygin, AP Kiselev, LI Goray, TM Zaboronkova, AY Kazakov & AS Kirpichnikova (eds), Proceedings of the International Conference Days on Diffraction 2020, DD 2020., 9274638, Proceedings of the International Conference Days on Diffraction 2020, DD 2020, Institute of Electrical and Electronics Engineers Inc., pp. 38-43, 2020 International Conference Days on Diffraction, DD 2020, St. Petersburg, Russian Federation, 25/05/20. https://doi.org/10.1109/DD49902.2020.9274638

APA

Ivanov, A. V. (2020). Exponential dichotomy of linear cocycles over irrational rotations. In O. V. Motygin, A. P. Kiselev, L. I. Goray, T. M. Zaboronkova, A. Y. Kazakov, & A. S. Kirpichnikova (Eds.), Proceedings of the International Conference Days on Diffraction 2020, DD 2020 (pp. 38-43). [9274638] (Proceedings of the International Conference Days on Diffraction 2020, DD 2020). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/DD49902.2020.9274638

Vancouver

Ivanov AV. Exponential dichotomy of linear cocycles over irrational rotations. In Motygin OV, Kiselev AP, Goray LI, Zaboronkova TM, Kazakov AY, Kirpichnikova AS, editors, Proceedings of the International Conference Days on Diffraction 2020, DD 2020. Institute of Electrical and Electronics Engineers Inc. 2020. p. 38-43. 9274638. (Proceedings of the International Conference Days on Diffraction 2020, DD 2020). https://doi.org/10.1109/DD49902.2020.9274638

Author

Ivanov, Alexey V. / Exponential dichotomy of linear cocycles over irrational rotations. Proceedings of the International Conference Days on Diffraction 2020, DD 2020. editor / O.V. Motygin ; A.P. Kiselev ; L.I. Goray ; T.M. Zaboronkova ; A.Ya. Kazakov ; A.S. Kirpichnikova. Institute of Electrical and Electronics Engineers Inc., 2020. pp. 38-43 (Proceedings of the International Conference Days on Diffraction 2020, DD 2020).

BibTeX

@inproceedings{3cbf18fed7b74105a97c9da31ab860ac,

title = "Exponential dichotomy of linear cocycles over irrational rotations",

abstract = "We study a linear cocycle over irrational rotation s?(x) = x+? of a circle T1. It is supposed that the cocycle is generated by a A? : T1 to SL(2, R) that depends on a small parameter ? « 1 and has the form of the Poincar{\'e} map corresponding to a singularly perturbed Schr{\"o}dinger equation. Under assumption that the eigenvalues of A?(x) are of the form exp (±?(x)/?), where ?(x) is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter ?. We show that in the limit ? ? 0 the cocycle exhibits ED for the most parameter values only if it is exponentially close to a constant cocycle. In the other case, when the cocycle is not close to a constant one and, thus, it does not possess ED, the Lyapunov exponent is typically large. ",

author = "Ivanov, {Alexey V.}",

note = "Publisher Copyright: {\textcopyright} 2020 IEEE. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.; 2020 International Conference Days on Diffraction, DD 2020 ; Conference date: 25-05-2020 Through 29-05-2020",

year = "2020",

month = may,

day = "25",

doi = "10.1109/DD49902.2020.9274638",

language = "English",

series = "Proceedings of the International Conference Days on Diffraction 2020, DD 2020",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

pages = "38--43",

editor = "O.V. Motygin and A.P. Kiselev and L.I. Goray and T.M. Zaboronkova and A.Ya. Kazakov and A.S. Kirpichnikova",

booktitle = "Proceedings of the International Conference Days on Diffraction 2020, DD 2020",

address = "United States",

url = "http://www.pdmi.ras.ru/~dd/download/DD20_program.pdf",

}

RIS

TY - GEN

T1 - Exponential dichotomy of linear cocycles over irrational rotations

AU - Ivanov, Alexey V.

PY - 2020/5/25

Y1 - 2020/5/25

N2 - We study a linear cocycle over irrational rotation s?(x) = x+? of a circle T1. It is supposed that the cocycle is generated by a A? : T1 to SL(2, R) that depends on a small parameter ? « 1 and has the form of the Poincaré map corresponding to a singularly perturbed Schrödinger equation. Under assumption that the eigenvalues of A?(x) are of the form exp (±?(x)/?), where ?(x) is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter ?. We show that in the limit ? ? 0 the cocycle exhibits ED for the most parameter values only if it is exponentially close to a constant cocycle. In the other case, when the cocycle is not close to a constant one and, thus, it does not possess ED, the Lyapunov exponent is typically large.

AB - We study a linear cocycle over irrational rotation s?(x) = x+? of a circle T1. It is supposed that the cocycle is generated by a A? : T1 to SL(2, R) that depends on a small parameter ? « 1 and has the form of the Poincaré map corresponding to a singularly perturbed Schrödinger equation. Under assumption that the eigenvalues of A?(x) are of the form exp (±?(x)/?), where ?(x) is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter ?. We show that in the limit ? ? 0 the cocycle exhibits ED for the most parameter values only if it is exponentially close to a constant cocycle. In the other case, when the cocycle is not close to a constant one and, thus, it does not possess ED, the Lyapunov exponent is typically large.

UR - http://www.scopus.com/inward/record.url?scp=85098964154&partnerID=8YFLogxK

U2 - 10.1109/DD49902.2020.9274638

DO - 10.1109/DD49902.2020.9274638

M3 - Conference contribution

AN - SCOPUS:85098964154

T3 - Proceedings of the International Conference Days on Diffraction 2020, DD 2020

SP - 38

EP - 43

BT - Proceedings of the International Conference Days on Diffraction 2020, DD 2020

A2 - Motygin, O.V.

A2 - Kiselev, A.P.

A2 - Goray, L.I.

A2 - Zaboronkova, T.M.

A2 - Kazakov, A.Ya.

A2 - Kirpichnikova, A.S.

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2020 International Conference Days on Diffraction, DD 2020

Y2 - 25 May 2020 through 29 May 2020

ER -

ID: 75583256