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 \hat{M} = M_T ∪ M_S. We use this algorithm to test the Marmi-Moussa-Yoccoz Conjecture about the Holder continuity of the function z → −iB(z) + logU(e^{2πiz}) on {z ∈ C : z ≥ 0}, where 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].