The 1/2--Complex Bruno function and the Yoccoz function. A numerical study of the Marmi--Moussa--Yoccoz Conjecture

Abstract

We study the 1/2--Complex Bruno function and we produce an algorithm to evaluate it numerically, giving a characterization of the monoid $\hat{\mathcal{M}}={\mathcal{M}}_T\cup {\mathcal{M}}_S$. We use this algorithm to test the Marmi--Moussa--Yoccoz Conjecture about the Hölder continuity of the function $z\mapsto -i\mathbf{B}(z)+\log U(e^{2\pi iz})$ on $\{z\in \mathbb{C}:\Im z\ge 0\}$, where $\mathbf{B}$ is the 1/2--complex Bruno function and $U$ is the Yoccoz function. We give a positive answer to an explicit question of S. Marmi et al [MMY2001].

21 pages, 11 figures, 2 tables