# New fractal set "Bubbles" and complex rotation numbers

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About 40 years ago Arnold suggested a construction of so called *complex rotation numbers*. Consider a circle diffeomorphism, lifted to a map of the real line that commutes with the shift by 1. For each complex number in the open upper half-plane consider the lifted map combined with the shift by this number. Glue the real line and the shifted real line via this map and factorize the cylinder obtained by the unit shift. Consider the modulus of the elliptic curve thus obtained. It is sometimes called the *complex rotation number* of the original map combined with the shift. We get the moduli map of the upper half plane into itself. The questions due to Arnold and Ghys can be formulated as follows:

*How the boundary values of the moduli map are related to the rotation number of the original shifted map considered as a function on the absolute?*

It appeared that for generic circle diffeomorphisms the boundary values of the moduli map form a fractal set ``Bubbles'' located inside the upper half plane: any rational point on the real axis belongs to a closed curve from this fractal set; this curve is called a bubble. The boundary values of the moduli map coincide with the rotation number when these values are irrational. The way of approximate calculation of bubbles is suggested, their possible bizarre shapes are drawn, and it is proved that the bubbles can intersect and self-intersect.

Self-similarity of the fractal set ``bubbles'' is established.

Recently it was found out that the fractal set ``bubbles'' *does not depend continuously* on the original circle diffeomorphism. This drastically distinguishes it from Arnold tongues that depend on the original diffeomorphism continuously.

Most part of these results belongs to N. Goncharuk; other part is obtained by Buff, Moldavskis, Rissler and others.