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

TL;DR

This paper investigates the structure of the natural extension of quadratic maps with irrational rotation numbers, revealing that most leaves are parabolic except for the Siegel disk, and discusses limitations related to Cremer points.

Contribution

It demonstrates that the regular part of the natural extension mostly consists of parabolic leaves, except for the invariant lift of the Siegel disk, and analyzes the impact of Cremer points on hyperbolic leaves.

Findings

Most leaves are parabolic except the Siegel disk lift.

Cremer points do not provide enough singularity for hyperbolic leaves.

The natural extension's irregular points form a continuum but lack hyperbolic leaves.

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.