Complex rotation numbers: bubbles and their intersections

TL;DR

This paper surveys the properties of complex rotation number bubbles related to circle diffeomorphisms, highlighting their intersections, self-intersections, and behaviors under perturbations, contributing to understanding their fractal structure.

Contribution

It provides a comprehensive survey of known properties of bubbles, introduces new insights on their intersections, and offers visualizations for perturbed M"obius diffeomorphisms.

Findings

Bubbles can intersect and self-intersect.

Approximate pictures of bubbles under perturbations.

Bubbles are a complex analogue to Arnold tongues.

Abstract

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 M\"obius circle diffeomorphisms.