Scaling Ratios and Triangles in Siegel Disks

1999/05/27 by Xavier Buff, Buff, Xavier, Christian Henriksen +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.math/9905173

13 pages, 13 PostScript figures

arxiv created 1999/05/27 · arxiv updated 2009/11/30

Abstract

Let f(z)=e^2iπθ z+z², where θ is a quadratic irrational. McMullen proved that the Siegel disk for f is self-similar about the critical point. We give a lower bound for the ratio of self-similarity, and we show that if θ=(√ 5-1)/2 is the golden mean, then there exists a triangle contained in the Siegel disk, and with one vertex at the critical point. This answers a 15 year old conjecture.

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