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.
"Complex rotation numbers: bubbles and their intersections." Anal. PDE 11 (7) 1787 - 1801, 2018. https://doi.org/10.2140/apde.2018.11.1787