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 M = M_T ∪ M_s . We use this algorithm to test the Marmi-Moussa-Yoccoz Conjecture about the Holder continuity of the function z ← −i B(z) + log U(e ² π iZ) on {z ∊ C : z ≥ O}, 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].

Keywords:

• Complex Bruno Function
• Yoccoz Function
• linearization of quadratic polynomial
• Littlewood-Paley dyadic decomposition
• continued fraction
• Farey series