## Institute for Mathematical Sciences

## Preprint ims99-4

** X. Buff and C. Henriksen**
* Scaling Ratios and Triangles in Siegel Disks*

Abstract: Let $f(z)=e^{2i\pi\theta} z+z^2$, where $\theta$ 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
$\theta=(\sqrt 5-1)/2$ 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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