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Funktsional. Anal. i Prilozhen., 2012, Volume 46, Issue 1, Pages 13–30 (Mi faa3064)  

This article is cited in 3 scientific papers (total in 3 papers)

Rotation Numbers and Moduli of Elliptic Curves

N. B. Goncharukab

a Independent University of Moscow
b M. V. Lomonosov Moscow State University

Abstract: Given a circle diffeomorphism $f$, we can construct a map taking each real number $a$ to the rotation number of the diffeomorphism $f+a$. In 1978, V. I. Arnold suggested a complex analog To this map. Given a complex number $z$ with $\operatorname{Im}z>0$, Arnold used the map $f+z$ to construct an elliptic curve. The moduli map takes every number $z$ to the modulus $\mu(z)$ of this elliptic curve.
In this article, we investigate the limit behaviour of the map $\mu$ in neighborhoods of the real intervals on which the rotation number of the diffeomorphism $f+a$ is rational. We show that the map $\mu$ extends analytically to any interior point of such an interval, excluding some finite set of exceptional points. Near exceptional points and the endpoints of the interval, the values of the function $\mu$ tend to the rotation number of the map $f+a$.
The union of the images of such intervals under the map $\mu$ is a fractal set in the upper half-plane. This fractal set is a complex analog to Arnold tongues.

Keywords: circle diffeomorphism, rotation number, elliptic curve, quasiconformal map

DOI: https://doi.org/10.4213/faa3064

Full text: PDF file (331 kB)
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English version:
Functional Analysis and Its Applications, 2012, 46:1, 11–25

Bibliographic databases:

UDC: 517.938
Received: 09.12.2010

Citation: N. B. Goncharuk, “Rotation Numbers and Moduli of Elliptic Curves”, Funktsional. Anal. i Prilozhen., 46:1 (2012), 13–30; Funct. Anal. Appl., 46:1 (2012), 11–25

Citation in format AMSBIB
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\by N.~B.~Goncharuk
\paper Rotation Numbers and Moduli of Elliptic Curves
\jour Funktsional. Anal. i Prilozhen.
\yr 2012
\vol 46
\issue 1
\pages 13--30
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\transl
\jour Funct. Anal. Appl.
\yr 2012
\vol 46
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84858436958}


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  • https://doi.org/10.4213/faa3064
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Buff X., Goncharuk N., “Complex Rotation Numbers”, J. Mod. Dyn., 9 (2015), 169–190  crossref  mathscinet  zmath  isi  scopus
    2. Goncharuk N., “Complex Rotation Numbers: Bubbles and Their Intersections”, Anal. PDE, 11:7 (2018), 1787–1801  crossref  mathscinet  zmath  isi  scopus
    3. Goncharuk N., “Self-Similarity of Bubbles”, Nonlinearity, 32:7 (2019), 2496–2521  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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