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Speiser meets Misiurewicz

Magnus Aspenberg, Weiwei Cui

Abstract

We propose a notion of Misiurewicz condition for transcendental entire functions and study perturbations of Speiser functions satisfying this condition in their parameter spaces (in the sense of Eremenko and Lyubich). We show that every Misiurewicz entire function can be approximated by hyperbolic maps in the same parameter space. Moreover, Misiurewicz functions are Lebesgue density points of hyperbolic maps if their Julia sets have zero Lebesgue measure. We also prove that the set of Misiurewicz Speiser functions has Lebesgue measure zero in the parameter space.

Speiser meets Misiurewicz

Abstract

We propose a notion of Misiurewicz condition for transcendental entire functions and study perturbations of Speiser functions satisfying this condition in their parameter spaces (in the sense of Eremenko and Lyubich). We show that every Misiurewicz entire function can be approximated by hyperbolic maps in the same parameter space. Moreover, Misiurewicz functions are Lebesgue density points of hyperbolic maps if their Julia sets have zero Lebesgue measure. We also prove that the set of Misiurewicz Speiser functions has Lebesgue measure zero in the parameter space.
Paper Structure (13 sections, 24 theorems, 106 equations, 1 figure)

This paper contains 13 sections, 24 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Some preliminaries
  3. Characterizing the Misiurewicz condition
  4. Parameter space of Speiser functions
  5. Speiser Misiurewicz maps are rare
  6. Phase-parameter relation
  7. Distortions along the post-singular set
  8. Proof of Theorem \ref{['main']}
  9. Perturbations of Misiurewicz maps
  10. Wiman-Valiron theory and growth in tracts
  11. An inducing scheme
  12. Misiurewicz and hyperbolicity
  13. Lebesgue measure of Misiurewicz Julia sets

Key Result

Theorem 1.1

Let $f\in\operatorname{\mathcal{S}}$ be Misiurewicz. Then $f$ can be approximated by hyperbolic maps in $\operatorname{\mathcal{M}}_f$.

Figures (1)

  • Figure 1: Constructing Misiurewicz maps with non-empty Fatou sets around a Misiurewicz map which has only asymptotic values. The Wiman-Valiron theory is used to create a sufficiently long iterate in logarithmic tracts which "swallows" all previous derivative growth.

Theorems & Definitions (47)

  • Definition 1.1
  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Remark 2.1
  • Definition 2.1
  • ...and 37 more