S Marmi
Complex Brjuno Functions
Abstract:
A Brjuno number is an irrational number such that a certain series involving its continued fraction convergents is finite. The set of Brjuno numbers is invariant under the modular group and on this set of points the arithmetical Brjuno function is bounded. This function is 1-periodic and satisfies a remarkable functional equation which expresses the fact that it is a co-cycle under the modular group action. I discuss several properties of this function and of its complexification which is obtained through the action of the modular group on the di-logarithm.
The Brjuno numbers and the Brjuno functions appear in the study of one-dimensional analytic small divisors problem.
Joint work with P. Moussa and J-Ch. Yoccoz
S. Marmi, P. Moussa and J-Ch. Yoccoz, Journal of the AMS 14 (2001) 783-841.
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