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Exponential iteration and Borel sets

David S. Lipham

Abstract

We determine the exact Borel class of the points whose iterates under $\exp(z)+a$ tend to infinity. We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.

Exponential iteration and Borel sets

Abstract

We determine the exact Borel class of the points whose iterates under tend to infinity. We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.

Paper Structure

This paper contains 14 sections, 13 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamics of entire functions
  4. Attracting and parabolic parameters
  5. Borel sets
  6. Baire category
  7. Lower semi-continuity
  8. A topological model of $\exp-1$
  9. The model
  10. Endpoints of $J(\mathcal{F})$
  11. Estimates of $t_{\underline s}$
  12. Stratifying the escaping endpoints
  13. Proof of Theorem 1
  14. Proof of Theorem 3

Key Result

Theorem 1

$I(f_a)$ is not $G_{\delta\sigma}$ for any $a\in \mathbb C$.

Figures (2)

  • Figure 1: Borel classes of $I(f_a)$ (in green only).
  • Figure 2: Disjoint curves in $I(f_{2+\frac{\pi}{2}i})$ which terminate at non-escaping points. See also rem2.

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Proposition 1
  • ...and 21 more