A new proof of a conjecture of Yoccoz
The authors give a new proof of a conjecture of Yoccoz, which provides an upper bound for the logarithm of the conformal radius of the Siegel disk of a map \(Q_\theta(z) = e^{2i\pi\theta} z + z^2\), in terms of Yoccoz's Brjuno function \(Y(\theta)\). Arithmetical properties of the rotation number \(\theta\) of a map of type \(f(z) = e^{2i\pi \theta} z + O(z^2)\) are intimately linked to the size of the Siegel disk of \(f\) at \(0\) (where the Siegel disk is the maximal domain containing 0 on which \(f\) is conjugated to a rotation). The authors' interesting proof follows Yoccoz's initial methods. Their result is then extended to other new families of polynomial maps like \(z\mapsto z^d+c\) when \(d>2\).
scientific article; zbMATH DE number 5899953
scientific article