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The \(1/2\)-complex Bruno function and the Yoccoz function: a numerical study of the Marmi-Moussa-Yoccoz conjecture. (English) Zbl 1173.37324

Summary: 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 \(z\mapsto -i\mathbb{B}(z)+ \log U\!\left(e^{2\pi i z}\right)\) on \(\{ z\in \mathbb{C}: \operatorname{Im} z \geq 0 \}\), where \(\mathbb{B}\) 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, P. Moussa and J.-C. Yoccoz [Commun. Math. Phys. 186, No. 2, 265–293 (1997; Zbl 0947.30018)].

MSC:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
11A55 Continued fractions
42B25 Maximal functions, Littlewood-Paley theory
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37M99 Approximation methods and numerical treatment of dynamical systems

Citations:

Zbl 0947.30018
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