Douady's conjecture on linearization asserts that a polynomial of polynomial degree at least two with an irrational
neutral fixed point has a Siegel disk there only when
its rotation number is a Brjuno number (Douady 1987,
p. 162). The converse is the Brjuno sufficiency theorem.
Dudko et al. (2026), using the reduction of Buff et al. (2026), proved this statement for all polynomials. Thus a polynomial
fixed point with multiplier , for irrational , is linearizable iff is a Brjuno number. The
analogous statement for rational functions of
degree at least two remains conjectural (Dudko et al. 2026).
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.Douady,
A. "Disques de Siegel et anneaux de Herman." Astérisque152-153,
151-172, 1987.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.
, for irrational 
, is linearizable iff 
is a Brjuno number. The
analogous statement for rational functions of
degree at least two remains conjectural (Dudko et al. 2026).
