A new proof of a conjecture of Yoccoz, Remarks, New results

Abstract

We give a new proof of the following conjecture of Yoccoz: the sum of the logarithm of the conformal radius of fixed Siegel disks of monic quadratic polynomials and of the Brjuno function of their rotation number is bounded from above. In a former article we obtained a first proof based on the control of parabolic explosion. Here, we present a more elementary proof based on Yoccoz's initial methods. We then extend this result to some new families of polynomials such as unicritical ones. We also show that the conjecture does not hold for some other families of polynomials.