Journal of Modern Dynamics (JMD)

Complex rotation numbers

Pages: 169 - 190, Volume 9, 2015      doi:10.3934/jmd.2015.9.169

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Xavier Buff - Institut deMathématiques de Toulouse, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France (email)
Nataliya Goncharuk - National Research University Higher School of Economics, Miasnitskaya Street 20, Moscow, Russia, and Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, Moscow, Russian Federation (email)

Abstract: We investigate the notion of complex rotation number which was introduced by V. I. Arnold in 1978. Let $f:\mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ be a (real) analytic orientation preserving circle diffeomorphism and let $\omega\in \mathbb{C}/\mathbb{Z}$ be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus {$z\in \mathbb{C}/\mathbb{Z} | 0 < Im(z)< Im(\omega)$} via the map $f+\omega$. This complex torus is isomorphic to $\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})$ for some appropriate $\tau\in \mathbb{C}/\mathbb{Z}$.
    According to Moldavskis [6], 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 [4], motivates us to introduce a new fractal set (``bubbles'') 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.

Keywords:  Complex tori, rotation numbers, diffeomorphisms of the circle, quasiconformal maps.
Mathematics Subject Classification:  Primary: 37E10; Secondary: 37E45, 30C62.

Received: July 2013;      Revised: May 2015;      Available Online: September 2015.