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

We study the 1/2–Complex Bruno function and we produce an algorithm to evaluate it numerically, giving a characterization of the monoid $\widehat{\mathcal{M}}={\mathcal{M}}_T\cup {\mathcal{M}}_S$. We use this algorithm to test the Marmi–Moussa–Yoccoz Conjecture about the H"older continuity of the function $z\mapsto -i\mathbf{B}(z)+\log U(e^{2\pi iz})$ on ${z\in $\mathbb{C}$: \Im z \geq 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].