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

In this note we show that the regular part of the natural extension (in the sense of Lyubich and Minsky) of quadratic map $f(z) = e^{2 πi θ}z + z^2$ with irrational $θ$ 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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