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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Complex Brjuno functions
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by Stefano Marmi, Pierre Moussa and Jean-Christophe Yoccoz
J. Amer. Math. Soc. 14 (2001), 783-841
Published electronically: May 30, 2001


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 $p/q\in {\mathbb {Q}}$.
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Bibliographic Information
  • Stefano Marmi
  • Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Loc. Rizzi, I-33100 Udine, Italy
  • Address at time of publication: Scuola Normale Superiore, Classe di Scienze, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
  • MR Author ID: 120140
  • Email:,
  • Pierre Moussa
  • Affiliation: Service de Physique Théorique, CEA/Saclay, 91191 Gif-Sur-Yvette, France
  • Email:
  • Jean-Christophe Yoccoz
  • Affiliation: Collège de France, 3 Rue d’Ulm, F-75005 Paris, France, and Université de Paris-Sud, Mathématiques, Batiment 425, F-91405 Orsay, France
  • Email:
  • Received by editor(s): February 16, 2000
  • Published electronically: May 30, 2001
  • Additional Notes: This work began during a visit of the first author at the S.Ph.T.–CEA/Saclay and at the Department of Mathematics of Orsay during the academic year 1993–1994. This research has been supported by the CNR, CNRS, INFN, MURST and an EEC grant

  • Dedicated: This paper is dedicated to Michael R. Herman
  • © Copyright 2001 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 14 (2001), 783-841
  • MSC (2000): Primary 37F50, 11A55, 32A40; Secondary 37F25, 46F15, 20G99
  • DOI:
  • MathSciNet review: 1839917