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Working paper

Complex rotation numbers

arxiv.org. math. Cornell University, 2013. No. 1308.3510.
Buff X., Goncharuk N. B.
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.