Exponential dichotomy of linear cocycles over irrational rotations

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

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 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.

Original languageEnglish
Title of host publicationProceedings of the International Conference Days on Diffraction 2020, DD 2020
EditorsO.V. Motygin, A.P. Kiselev, L.I. Goray, T.M. Zaboronkova, A.Ya. Kazakov, A.S. Kirpichnikova
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages38-43
Number of pages6
ISBN (Electronic)9780738142791
DOIs
StatePublished - 25 May 2020
Event2020 International Conference Days on Diffraction, DD 2020 - ПОМИ РАН, St. Petersburg, Russian Federation
Duration: 25 May 202029 May 2020
http://www.pdmi.ras.ru/~dd/download/DD20_program.pdf

Publication series

NameProceedings of the International Conference Days on Diffraction 2020, DD 2020

Conference

Conference2020 International Conference Days on Diffraction, DD 2020
CountryRussian Federation
CitySt. Petersburg
Period25/05/2029/05/20
Internet address

Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Acoustics and Ultrasonics
  • Instrumentation
  • Atomic and Molecular Physics, and Optics
  • Radiation

Fingerprint Dive into the research topics of 'Exponential dichotomy of linear cocycles over irrational rotations'. Together they form a unique fingerprint.

Cite this