American Institute of Mathematical Sciences

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

Complex rotation numbers

 1 Institut deMathématiques de Toulouse, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France 2 National Research University Higher School of Economics, Miasnitskaya Street 20, Moscow, Russia, and Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, Moscow, Russian Federation

Received  July 2013 Revised  May 2015 Published  September 2015

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.
Citation: Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169
References:
 [1] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations,, Grund-lehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], (1983).   Google Scholar [2] É. Ghys, Groups Acting on the Circle: A Selection of Open Problems,, Opening Lecture of Spring School, (2008).   Google Scholar [3] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: Three theorems of J.-C. Yoccoz,, in Topological Methods in Modern Mathematics (Stony Brook, (1991), 467.   Google Scholar [4] N. B. Goncharuk, Rotation numbers and moduli of elliptic curves,, Funct. Anal. Appl., 46 (2012), 11.  doi: 10.1007/s10688-012-0002-8.  Google Scholar [5] Y. Ilyashenko and V. Moldavskis, Morse-Smale circle diffeomorphisms and moduli of elliptic curves,, Mosc. Math. J., 3 (2003), 531.   Google Scholar [6] V. S. Moldavskiĭ, Moduli of elliptic curves and rotation numbers of diffeomorphisms of the circle,, Funct. Anal. Appl., 35 (2001), 234.  doi: 10.1023/A:1012391215252.  Google Scholar [7] E. Risler, Linéarisation des perturbations holomorphes des rotations et applications,, Mém. Soc. Math. Fr. (N.S.), (1999).   Google Scholar [8] M. Tsujii, Rotation number and one-parameter families of circle diffeomorphisms,, Ergodic Theory Dynam. Systems, 12 (1992), 359.  doi: 10.1017/S0143385700006805.  Google Scholar [9] J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,, Ann. Sci. École Norm. Sup. (4), 17 (1984), 333.   Google Scholar

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References:
 [1] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations,, Grund-lehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], (1983).   Google Scholar [2] É. Ghys, Groups Acting on the Circle: A Selection of Open Problems,, Opening Lecture of Spring School, (2008).   Google Scholar [3] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: Three theorems of J.-C. Yoccoz,, in Topological Methods in Modern Mathematics (Stony Brook, (1991), 467.   Google Scholar [4] N. B. Goncharuk, Rotation numbers and moduli of elliptic curves,, Funct. Anal. Appl., 46 (2012), 11.  doi: 10.1007/s10688-012-0002-8.  Google Scholar [5] Y. Ilyashenko and V. Moldavskis, Morse-Smale circle diffeomorphisms and moduli of elliptic curves,, Mosc. Math. J., 3 (2003), 531.   Google Scholar [6] V. S. Moldavskiĭ, Moduli of elliptic curves and rotation numbers of diffeomorphisms of the circle,, Funct. Anal. Appl., 35 (2001), 234.  doi: 10.1023/A:1012391215252.  Google Scholar [7] E. Risler, Linéarisation des perturbations holomorphes des rotations et applications,, Mém. Soc. Math. Fr. (N.S.), (1999).   Google Scholar [8] M. Tsujii, Rotation number and one-parameter families of circle diffeomorphisms,, Ergodic Theory Dynam. Systems, 12 (1992), 359.  doi: 10.1017/S0143385700006805.  Google Scholar [9] J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,, Ann. Sci. École Norm. Sup. (4), 17 (1984), 333.   Google Scholar
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