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Rotation numbers and moduli of elliptic curves

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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 Im z > 0, Arnold used the map f + z to construct an elliptic curve. The moduli map takes every number z to the modulus µ(z) of this elliptic curve. In this article, we investigate the limit behaviour of the map µ in neighborhoods of the real intervals on which the rotation number of the diffeomorphism f + a is rational. We show that the map µ 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 µ tend to the rotation number of the map f + a. The union of the images of such intervals under the map µ is a fractal set in the upper half-plane. This fractal set is a complex analog to Arnold tongues.

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... The following theorem is proved in [4]; the proof is based on previous results by E. Risler, V. Moldavskij, Yu. Ilyashenko, and N. Goncharuk [12,11,8,5]). ...
... This construction was proposed by X. Buff; see [5,4] for more details. The key idea of this construction is contained in [12], but there it was used in different circumstances. ...
... For rot f = p q , the construction of γ should be slightly modified: φ j are linearizing charts of f q at its fixed points, γ j are arcs of circles in charts φ j , γ = γ j , γ winds above repelling periodic points of f and below attracting periodic points of f , and we choose γ j so that f (γ) is above γ in C/Z. The rest of the construction is analogous to the case of rot f = 0. Theorem 4 ( [5]; see also [4,Sec. 6]). ...
Preprint
The construction of complex rotation numbers, due to V.Arnold, gives rise to a fractal-like set "bubbles" related to a circle diffeomorphism. "Bubbles" is a complex analogue to Arnold tongues. This article contains a survey of the known properties of bubbles, as well as a variety of open questions. In particular, we show that bubbles can intersect and self-intersect, and provide approximate pictures of bubbles for perturbations of M\"obius circle diffeomorphisms.
... Two independent proofs of this conjecture were given in [15] and [14]. This statement does not hold if f + ω 0 is hyperbolic, as was proved in [9]; this result was strengthened in [7]. The case of a diffeomorphism with parabolic cycles was studied by Lacroix (unpublished) and in [7]. ...
... This statement does not hold if f + ω 0 is hyperbolic, as was proved in [9]; this result was strengthened in [7]. The case of a diffeomorphism with parabolic cycles was studied by Lacroix (unpublished) and in [7]. ...
... Namely, we will construct a non-degenerate torus E(g) with modulus τ (G). The construction was suggested by Buff and used in [4,7]. ...
Article
Full-text available
Bubbles is a fractal-like set related to a circle diffeomorphism; they are a complex analogue to Arnold tongues. In this article, we prove an approximate self-similarity of bubbles.
... Two independent proofs of this conjecture were given in [11] and [10]. This statement does not hold if f + ω 0 is hyperbolic, as was proved in [6]; this result was strengthened in [5]. The case of a diffeomorphism with parabolic cycles was studied by J.Lacroix (unpublished) and in [5]. ...
... This statement does not hold if f + ω 0 is hyperbolic, as was proved in [6]; this result was strengthened in [5]. The case of a diffeomorphism with parabolic cycles was studied by J.Lacroix (unpublished) and in [5]. ...
... The following result gives the description of the limit behaviour of τ f near the real line. The analyticity ofτ f in the last subcase below is not included in [2,Main theorem], but it constitutes [5,Theorem 1.2], see also [2,Theorem 2]. ...
Preprint
Bubbles is a fractal-like set related to a circle diffeomorphism; they are a complex analogue to Arnold tongues. In this article, we prove an approximate self-similarity of bubbles.
... The following theorem is proved in [4]; the proof is based on previous results by E. Risler, V. Moldavskij, Yu. Ilyashenko, and N. Goncharuk [12,11,8,5]). ...
... This construction was proposed by X. Buff; see [5,4] for more details. The key idea of this construction is contained in [12], but there it was used in different circumstances. ...
... For rot f = p q , the construction of γ should be slightly modified: φ j are linearizing charts of f q at its fixed points, γ j are arcs of circles in charts φ j , γ = γ j , γ winds above repelling periodic points of f and below attracting periodic points of f , and we choose γ j so that f (γ) is above γ in C/Z. The rest of the construction is analogous to the case of rot f = 0. Theorem 4 ( [5]; see also [4,Sec. 6]). ...
Article
Full-text available
The construction of complex rotation numbers, due to V.Arnold, gives rise to a fractal-like set "bubbles" related to a circle diffeomorphism. "Bubbles" is a complex analogue to Arnold tongues. This article contains a survey of the known properties of bubbles, as well as a variety of open questions. In particular, we show that bubbles can intersect and self-intersect, and provide approximate pictures of bubbles for perturbations of M\"obius circle diffeomorphisms.
... The case of Diophantine rotation numbers was investigated earlier by E.Risler [6,Chapter 2] and V.Moldavskis [5] independently. The case of parabolic cycles was studied by J.Lacroix (unpublished) and N.Goncharuk [3] independently. The case of hyperbolic diffeomorphisms was dealt first by Ilyashenko and Moldavskis [4], then this result was improved by N.Goncharuk [3]. ...
... The case of parabolic cycles was studied by J.Lacroix (unpublished) and N.Goncharuk [3] independently. The case of hyperbolic diffeomorphisms was dealt first by Ilyashenko and Moldavskis [4], then this result was improved by N.Goncharuk [3]. For exact statements of these results, see Section 2. ...
... Due to the Main Theorem, this set contains R/Z and a countable number of loops -"bubbles", the endpoints of bubbles are rational points of R/Z (see the sketch at Fig. 1). Due to [3], these loops are analytic curves. ...
Article
Full-text available
We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z \to R/Z be an orientation preserving circle diffeomorphism and let {\omega} \in C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus {z \in C/Z | 0< Im(z)< Im({\omega})} via the map f+{\omega}. This complex torus is isomorphic to C/(Z+{\tau} Z) for some appropriate {\tau} \in C/Z. According to Moldavskis (2001), if the ordinary rotation number rot (f+\omega_0) is Diophantine and if {\omega} tends to \omega_0 non tangentially to the real axis, then {\tau} tends to rot (f+\omega_0). We show that the Diophantine and non tangential assumptions are unnecessary: if rot (f+\omega_0) is irrational then {\tau} tends to rot (f+\omega_0) as {\omega} tends to \omega_0. This, together with results of N.Goncharuk (2012), motivates us to introduce a new fractal set, given by the limit values of {\tau} as {\omega} tends to the real axis. For the rational values of rot (f+\omega_0), these limits do not necessarily coincide with rot (f+\omega_0)$ and form a countable number of analytic loops in the upper half-plane.
Article
To any circle diffeomorphism there corresponds, by a classical construction of V. I. Arnold, a one-parameter family of elliptic curves. Arnold conjectured that, as the parameter approaches zero, the modulus of the corresponding elliptic curve tends to the (Diophantine) rotation number of the original diffeomorphism. In this paper, we disprove the generalization of this conjecture to the case when the diffeomorphism in question is Morse-Smale. The proof relies on the theory of quasiconformal mappings.
Moduli of elliptic curves and rotation numbers of circle diffeomorphisms English transl.: Functional Anal
  • V S Moldavskii
  • V S Moldavskii
¨ Uber iterierte Funktionen
  • E Schrö
E. Schrö, " ¨ Uber iterierte Funktionen, " Math. Ann., 3 (1871), 296–322.
Lectures on Quasiconformal Mappings, Van Nostrand Mathematical Studies
  • L Ahlfors
Über iterierte Funktionen
  • E Schröder
  • E. Schröder