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Scaling Ratios and Triangles in Siegel Disks

Xavier Buff·Christian Henriksen
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Abstract

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.

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Xavier Buff, Christian Henriksen. 1999-05-27. Scaling Ratios and Triangles in Siegel Disks. https://arxiv.org/abs/math/9905173

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