Scaling Ratios and Triangles in Siegel Disks
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Let $f(z)=e^{2iπθ} z+z^2$, 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 $θ=(\sqrt 5-1)/2$ 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.
13 pages, 13 PostScript figures
13 pages, 13 PostScript figures