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## Complex rotation numbers

Journal of Modern Dynamics. 2015. Vol. 9. P. 169-190.
Buff X., Goncharuk Nataliya.

We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z ->  R/Z be a (real) analytic 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 V.Moldavskis, if the ordinary rotation number rot(f+omega0) is Diophantine and if omega tends to omega0 non tangentially to the real axis, then tau tends to rot(f+omega0). We show that the Diophatine and non tangential assumptions are unnecessary: if rot(f+omega0) is irrational then tau tends to rot(f+omega0) as omega tends to omega0.  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+omega0), these limits do not necessarily coincide with rot(f+omega0) and form a countable number of analytic loops in the upper half-plane.