Vol. 11, No. 7, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Complex rotation numbers: bubbles and their intersections

Nataliya Goncharuk

Vol. 11 (2018), No. 7, 1787–1801
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 Möbius circle diffeomorphisms.

Keywords
complex tori, rotation numbers, diffeomorphisms of the circle
Mathematical Subject Classification 2010
Primary: 37E10, 37E45
Milestones
Received: 3 August 2017
Revised: 8 December 2017
Accepted: 9 April 2018
Published: 20 May 2018
Authors
Nataliya Goncharuk
Department of Mathematics
Cornell University
Ithaca, NY
United States