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 {\small $\hat{\mathcal{M}}=\mathcal{M}_T\cup \mathcal{M}_S$}. We use this algorithm to test the Marmi-Moussa-Yoccoz Conjecture about the Holder continuity of the function {\small $z\mapsto -i\B(z)+ \log U\!\left(e^{2\pi i z}\right)$} on {\small $\{ z\in \C: \Im z \geq 0 \}$}, where {\small $\B$} 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 [Marmi et al. 01].