Complex rotation numbers

Abstract

We investigate the notion of complex rotation number which was introduced by V.

Complex rotation numbers: bubbles and their intersections

“…The answer to this question is Yes, and this holomorphic function is the complex rotation number τ (f +ω). The following theorem is proved in [4]; the proof is based on previous results by E. Risler, V. Moldavskij, Yu. Ilyashenko, and N. Goncharuk [12,11,8,5]).…”

“…Theorem 3 (X. Buff, N. Goncharuk [4]). Let f : R/Z → R/Z be an orientationpreserving analytic circle diffeomorphism.…”

“…For non-hyperbolic f 1 , f 2 , their complex rotation numbers coincide with rotation numbers, so this lemma trivially repeats the invariance of rotation numbers under conjugacies. For hyperbolic diffeomorphisms, the proof of this lemma is implicitly contained in [4], see also Section 5 below.…”

“…In [4,Main Theorem] the authors prove that the p/q-bubble (with coprime p, q) is within a disc of radius D f /(4πq 2 ) tangent to R/Z at p/q, where D f is the distortion…”

“…This construction was proposed by X. Buff; see [5,4] for more details. The key idea of this construction is contained in [12], but there it was used in different circumstances.…”