Diophantine Conditions and Real or Complex Brjuno Functions

  • Pierre Moussa
  • Stefano Marmi
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 550)


The continued fraction expansion of the real number x=a 0+x 0, a 0∈ℤ is given by 0≤x n<1, x n −1 =a n+1+x n+1+a n+1∈ℕ for n≥0. The Brjuno function is then \( B(x) = \sum\nolimits_{n = 0}^\infty {x_0 x_1 \ldots x_{n - 1} \ln (x_n^{ - 1} )} \) and the number x satisfies the Brjuno diophantine condition whenever B(x) is bounded. Invariant circles under a complex rotation persist when the map is analytically perturbed, if and only if the rotation number satisfies the Brjuno condition, and the same holds for invariant circles in the semi-standard and standard map cases. In this lecture, we will review some properties of the Brjuno function, and give some generalisations related to familiar diophantine conditions. The Brjuno function is highly singular and takes value +∞ on a dense set including rationals. We present a regularisation leading to a complex function holomorphic in the upper half plane. Its imaginary part tends to the Brjuno function on the real axis, the real part remaining bounded, and we also indicate its transformation under the modular group.


Half Plane Rotation Number Modular Group Continue Fraction Expansion Invariant Circle 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Pierre Moussa
    • 1
  • Stefano Marmi
    • 2
  1. 1.Service de Physique ThéoriqueCEA/SaclayGif sur Yvette cedexFrance
  2. 2.Dipartimento di Matematica“U.Dini” Università di FirenzeFirenzeItaly

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