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

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.

11 pages, 1 figure

References in corpus (2)

• A note on hyperbolic leaves and wild laminations of rational functions
• Quadratic Julia Sets with Positive Area