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Brjuno Number

A Brjuno number is an irrational number alpha whose continued fraction convergents p_n/q_n satisfy

sum_(n=0)^infty(lnq_(n+1))/(q_n)<infty

(Marmi et al. 1997, 2001). Brjuno numbers arise in the study of one-dimensional analytic small divisors problems, and Brjuno (1971, 1972) proved that all "germs" with linear part lambda=e^(2piialpha) are linearizable if alpha is a Brjuno number. Yoccoz (1995) proved necessity for the quadratic polynomial e^(2piialpha)z+z^2. This means that the Brjuno condition characterizes rotation numbers for which every analytic germ with that multiplier is linearizable. It does not assert necessity for every analytic germ individually.

For polynomials, the condition is also necessary at every irrational neutral fixed point. Dudko et al. (2026), together with the reduction of Buff et al. (2026), proved that a polynomial of polynomial degree at least two with multiplier e^(2piialpha) has a Siegel disk at the fixed point iff alpha is a Brjuno number.

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References

Brjuno, A. D. "Analytical Form of Differential Equations." Trans. Moscow Math. Soc. 25, 131-288, 1971.Brjuno, A. D. "Analytical Form of Differential Equations. II." Trans. Moscow Math. Soc. 26, 199-239, 1972.Buff, X.; Chéritat, A.; and Roesch, P. "Biggest Bounded Type Siegel Disks of Monic Polynomials Include Those That Stick to All Critical Points." 3 Oct 2026. https://arxiv.org/abs/2606.24833.Dudko, D.; Luo, Y.; and Zhang, R. "Multicritical Pseudo-Siegel Bounds and the Optimality of the Brjuno Condition for All Neutral Polynomials." 27 Oct 2026. https://arxiv.org/abs/2610.37468.Marmi, S.; Moussa, P.; and Yoccoz, J.-C. "The Brjuno Functions and Their Regularity Properties." Comm. Math. Phys. 186, 265-293, 1997.Marmi, S.; Moussa, P.; and Yoccoz, J.-C. "Complex Brjuno Functions." J. Amer. Math. Soc. 14, 783-841, 2001.Siegel, C. L. "Iteration of Analytic Functions." Ann. Math. 43, 807-812, 1942.Yoccoz, J.-C. "Théorème de Siegel, nombres de Bruno et polynômes quadratiques." Astérique 231, 3-88, 1995.

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Brjuno Number

Cite this as:

Weisstein, Eric W. "Brjuno Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BrjunoNumber.html

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