2003/05/05 by Xavier Buff, Buff, Xavier, Arnaud Chéritat +2 · 1 citation
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS
paper · pdf · doi:10.48550/arxiv.math/0305080
22 pages, 4 figures
arxiv created 2003/05/05 · openalex publication_date 2003/05/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If \a is an irrational number, we let \pn/qn\n≥ 0, be the approximants given by its continued fraction expansion. The Bruno series B(\a) is defined as B(\a)=∑n≥ 0 \fraclog qn+1qn. The quadratic polynomial P_\a:z↦ e2iπ\az+z2 has an indifferent fixed point at the origin. If P_\a is linearizable, we let r(\a) be the conformal radius of the Siegel disk and we set r(\a)=0 otherwise. Yoccoz proved that if B(\a)=∞, then r(\a)=0 and P_\a is not linearizable. In this article, we present a different proof and we show that there exists a constant C such that for all irrational number \a with B(\a)