The construction of complex rotation numbers, due to V. Arnold, gives rise to a
fractal-like set “bubbles” related to a circle diffeomorphism. “Bubbles” is a complex
analogue to Arnold tongues.
This article contains a survey of the known properties of bubbles, as well as a
variety of open questions. In particular, we show that bubbles can intersect and
self-intersect, and provide approximate pictures of bubbles for perturbations of
Möbius circle diffeomorphisms.
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