Complex rotation numbers: bubbles and their intersections

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.2010 Mathematics Subject Classification. 37E10, 37E45.

Self-similarity of bubbles

“…There is no canonical choice of an analytic chart on the circle [x, g(x)]/g. However the complex rotation number does not depend on the choice of the analytic chart, due to the following lemma (for the proof, see [4,Lemma 8] or Remark 37 below).…”

“…Each monotonic family of circle diffeomorphisms f ω gives rise to the "fractal-like" set p/q∈Q B p/q,Fω (bubbles) in H, containing countably many analytic curves "growing" from rational points. These curves may intersect and self-intersect as shown in [4]; some pictures of bubbles are also presented there. The aim of this article is to prove that the bubbles of monotonic families are approximately self-similar near rational points, and to describe the "limit shapes" of bubbles.…”

“…Let K f := K be the transition map that corresponds to the parabolic diffeomorphism f . Recall that the additional assumption on K in Theorem 8 was as follows: (4) "For any c, whenever rot (K f + c) = p/q, the map K f + c has at most one parabolic orbit".…”

“…This result justifies the term "complex rotation number" for τ F . The limit behavior of τ F on the real axis for non-Diophantine rotation numbers was further studied in a sequence of papers [6], [7], [2], [3], [4]. In particular, we have the following result: Theorem 3 ( [2]).…”

“…For the proof of Lemma 5, see [3][ Lemma 8]. Recall that a circle diffeomorphism is called hyperbolic if it has a rational rotation number and the multipliers of all its periodic orbits are not equal to ±1.…”