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arXiv · 1103.2905

On the natural extension of a map with a Siegel or Cremer point

Carlos Cabrera·Tomoki Kawahira
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Abstract

In this note we show that the regular part of the natural extension (in the sense of Lyubich and Minsky) of quadratic map $f(z) = e^{2 \pi i \theta}z + z^2$ with irrational $\theta$ of bounded type has only parabolic leaves except the invariant lift of the Siegel disk. We also show that though the natural extension of a rational function with a Cremer fixed point has a continuum of irregular points, it can not supply enough singularity to apply the Gross star theorem to find hyperbolic leaves.

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Carlos Cabrera, Tomoki Kawahira. 2011-03-15. On the natural extension of a map with a Siegel or Cremer point. https://arxiv.org/abs/1103.2905

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