Complex rotation numbers: bubbles and their intersections

Goncharuk, Nataliya

doi:10.2140/apde.2018.11.1787

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Complex rotation numbers: bubbles and their intersections

Nataliya Goncharuk

Vol. 11 (2018), No. 7, 1787–1801

DOI: 10.2140/apde.2018.11.1787

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

Vol. 11, No. 7, 2018