A proof of the Marmi–Moussa–Yoccoz conjecture for rotation numbers of high type

Abstract

Abstract. -Marmi Moussa and Yoccoz conjectured that some error function Υ, related to the approximation of the size of Siegel disk by some arithmetic function of the rotation number θ, is a Hölder continuous function of θ with exponent 1/2. Using the renormalization invariant class of Inou and Shishikura, we prove this conjecture for the restriction of Υ to a class of high type numbers.

Multifractal analysis of the Brjuno function

“…The Marmi-MoussaYoccoz conjecture is another regularity problem related with the Brjuno function: It states that the sum of B and the logarithm of the conformal radius of the Siegel disk of a monic quadratic polynomial is C 1/2 , see [23] p. 267. Key steps towards its resolution have been obtained by X. Buff, D. Cheraghi and A. Chéritat, see [9,14]. Local properties of B were recently investigated by M. Balazard and B. Martin in [3]: They showed that its Lebesgue points are precisely the Bruno numbers, and they obtained precise estimates of the average of B over an interval, which will play a key-role in our study, see e.g (9).…”

“…Let x 0 ∈ R. First note that, if x 0 ∈ Q, then the result follows from (10). If x 0 / ∈ Q, we apply (14) to the sequence r n = p n q n of convergents of x 0 . We now pick h n = ε/q 2 n , where ε is positive and such thatCε log(1/ε) ≤ 1/4 (whereC is the constant in Lemma 2).…”

One hundred years of complex dynamics

“…The nature of the continued fraction expansion of ω was shown by Yoccoz [61,62] to determine whether the corresponding periodic point is in a Siegel disc or is a Cremer point: it was shown that a necessary and sufficient condition for a Siegel disc is that ω is a Bryuno number. (A conjecture in that paper was recently proved by Cheraghi & Chéritat [63].) In another result concerning the type of the number, Herman had proved [64] that if ω satisfies a Diophantine condition, then any periodic orbit of Siegel domain boundaries contains a critical point.…”

Near parabolic renormalization for unicritical holomorphic maps

“…It allows to study the hedgehogs and the size of Siegel disks. In a recent preprint, [CC15] proved the Marmi Moussa Yoccoz conjecture restricted to high type numbers. Cheraghi has given many other applications [Che10,Che13,AC12].…”