Parabolic explosion and the size of Siegel disks in the quadratic family

Abstract

For an irrational number θ, let r(θ) be the conformal radius at f=0 of the polynomial z ↦ ei2πθz + θ2. Let Y be Yoccoz’s arithmetic Brjuno function. The linearization theorem of Yoccoz implies that there exists a real constant C1 such that for all θ irrational with Y(θ) ≤ +∞,

C1 ≤ Y(θ) + log(r(θ)).

Yoccoz also proved a similar but weaker inequality on the other side, which in particular implies non linearizability of the polynomial when Y(θ) = +∞. The author found a new and independent proof of non linearizability. One of the tools of this new proof is the control on parabolic explosion. In a joint work with X. Buff, we pushed this further and were able to prove the existence of a real constant C2 such that for all θ with r(θ) > 0,

Y(θ) + log(r(θ)) ≤ C2,

enhancing Yoccoz’s second inequality.