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Slowly recurrent Collet-Eckmann maps with non-empty Fatou set

Magnus Aspenberg, Mats Bylund, Weiwei Cui

Abstract

In this paper we study rational Collet-Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, they are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree $d \geq 2$.

Slowly recurrent Collet-Eckmann maps with non-empty Fatou set

Abstract

In this paper we study rational Collet-Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, they are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree .
Paper Structure (14 sections, 17 theorems, 131 equations)

This paper contains 14 sections, 17 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas
  4. Phase-parameter relations
  5. Transversality
  6. Weak parameter dependence
  7. Distortion estimate
  8. Expansion during bound periods
  9. Expansion during free periods
  10. At the next free return
  11. Main distortion lemma (MDL)
  12. Consequences of MDL
  13. Large deviations and escape of partition elements
  14. Conclusion and proof

Key Result

Theorem 1.3

Any slowly recurrent rational Collet--Eckmann map $f \in \Lambda_{d,\overline{p'}}$ of degree $d \geq 2$, for which the Julia set is not the entire sphere, is a Lebesgue density point of hyperbolic maps in $\Lambda_{d,\overline{p'}}$.

Theorems & Definitions (39)

  • Definition 1.1: Collet--Eckmann condition
  • Definition 1.2: Slow recurrence
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2: Partition element
  • Definition 2.3: Bound period for parameters
  • Definition 2.4: Bound period for partition elements
  • Definition 2.5
  • Lemma 3.1
  • ...and 29 more