(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
are linearizable if
is a Brjuno number. Yoccoz (1995) proved that this condition is also necessary.
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.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." Astérique231,
3-88, 1995.
be the sequence of convergents
of the continued fraction of a number 
. Then a Brjuno number is an irrational
number such that
are linearizable if 
is a Brjuno number. Yoccoz (1995) proved that this condition is also necessary.
