Complex Brjuno functions

Marmi, Stefano^1, 2 ; Moussa, Pierre³ ; Yoccoz, Jean-Christophe⁴

¹ Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Loc. Rizzi, I-33100 Udine, Italy
² Scuola Normale Superiore, Classe di Scienze, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
³ Service de Physique Théorique, CEA/Saclay, 91191 Gif-Sur-Yvette, France
⁴ Collège de France, 3 Rue d’Ulm, F-75005 Paris, France, and Université de Paris-Sud, Mathématiques, Batiment 425, F-91405 Orsay, France

Journal of the American Mathematical Society, Tome 14 (2001) no. 4, pp. 783-841

Résumé

The Brjuno function arises naturally in the study of analytic small divisors problems in one dimension. It belongs to $\hbox {BMO}({\mathbb {T}}^{1})$ and it is stable under Hölder perturbations. It is related to the size of Siegel disks by various rigorous and conjectural results. In this work we show how to extend the Brjuno function to a holomorphic function on ${\mathbb {H}}/{\mathbb {Z}}$, the complex Brjuno function. This has an explicit expression in terms of a series of transformed dilogarithms under the action of the modular group. The extension is obtained using a complex analogue of the continued fraction expansion of a real number. Since our method is based on the use of hyperfunctions, it applies to less regular functions than the Brjuno function and it is quite general. We prove that the harmonic conjugate of the Brjuno function is bounded. Its trace on ${\mathbb {R}}/{\mathbb {Z}}$ is continuous at all irrational points and has a jump of $\pi /q$ at each rational point $p/q\in {\mathbb {Q}}$.

Détail

DOI : 10.1090/S0894-0347-01-00371-X

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