The Brjuno Functions and Their Regularity Properties

Multifractal analysis of the Brjuno function

“…First, B is one example of a family B α introduced by J.-C. Yoccoz in [35], and further studied in [22,23]: In the definition of B, the usual continued fraction algorithm is replaced by α-continued fractions expansions, see [27]. A similar analysis as the one that we performed could be developed for B α .…”

“…Marmi, P. Moussa and J.-C. Yoccoz determined the optimal global regularity of B, showing that it belongs to BMO, see [23]. 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].…”

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

“…In the arithmetic part, we deduce Hölder continuity of an extension of Υ to IS, using boundedness of Υ and applying the renormalization operator infinitely many times. This argument is very much in the spirit of [MMY97], and involves arithmetic estimates on the continued fraction that we reproduce here.…”

“…The connection with our work starts with the discussion in [MMY97]. In their notations, what we call α they call x and they use α as a parameter for a family of continued fraction algorithms.…”

“…In [MMY97] (Theorem 4.6), it was proven that B 1 − B 1/2 is Hölder-continuous with exponent 1/2. It follows that the main theorem of the present article is independent of the choice of the continued fraction expansion, B = B 1 or B = B 1/2 .…”