The Brjuno function continuously estimates the size of quadratic Siegel disks

Abstract

If α is an irrational number, Yoccoz defined the Brjuno function Φ by Φ(α) = ∑n≥0 α0α 1 ⋯ αn-1 log 1/αn, where α0 is the fractional part of α and αn+1 is the fractional part of l/αn. The numbers α such that Φ(α)