A Siegel disk is a periodic connected component of the Fatou set on which the first-return iterate is analytically conjugate to an irrational
rotation of a disk. At an irrational neutral fixed point moved to 0, this means that
a holomorphic change of coordinates conjugates 
to 
for an irrational number 
(Siegel 1942, Yoccoz 1995).
A Brjuno number satisfies an arithmetic condition on the denominators of its continued fraction convergents that guarantees such linearization. Dudko
et al. (2026), using a reduction of Buff et al. (2026), proved that
for every polynomial 
of polynomial degree
at least two with 
and 
for an irrational
number 
,
the fixed point has a Siegel disk iff 
is a Brjuno number. This
proves the polynomial case of Douady's conjecture
on the optimality of the Brjuno condition. Neutral periodic points reduce to this
case by iteration.
