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Indecomposable continua and the Julia sets of polynomial-like mappings

Elena Gomes

Abstract

Let $f$ be a polynomial-like mapping of the sphere of degree $d \geq 2$. We show that the Julia set $J(f)$ of $f$ cannot be the union of a finite number of proper indecomposable subcontinua. As a corollary, we prove that $J(f)$ is an indecomposable continuum if and only if there exists a prime end of some complementary region of $J(f)$ whose impression is the entire $J(f)$, generalizing a result by Childers, Mayer and Rogers.

Indecomposable continua and the Julia sets of polynomial-like mappings

Abstract

Let be a polynomial-like mapping of the sphere of degree . We show that the Julia set of cannot be the union of a finite number of proper indecomposable subcontinua. As a corollary, we prove that is an indecomposable continuum if and only if there exists a prime end of some complementary region of whose impression is the entire , generalizing a result by Childers, Mayer and Rogers.
Paper Structure (17 sections, 27 theorems, 9 equations, 3 figures)

This paper contains 17 sections, 27 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Acknowledgements
  4. Preliminaries
  5. Indecomposable continua
  6. Polynomial-like mappings
  7. Proof of Theorem \ref{['teo J']}
  8. A theorem regarding internal composants
  9. Composants
  10. Proof of Theorem \ref{['teo K']}
  11. Indecomposable subcontinua of Julia sets
  12. Invariance of the decomposition
  13. Proof of Lemma \ref{['lema preimagen de B-A es conexo']}
  14. Proof of Lemma \ref{['lema L']}
  15. Prime ends and indecomposability of the Julia set
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $f : S^2 \to S^2$ be a polynomial-like mapping of degree $d \geq 2$. Then its Julia set $J(f)$ is not the union of finitely many proper indecomposable subcontinua.

Figures (3)

  • Figure 1: An illustration of Theorem \ref{['teo krasi']}.
  • Figure 2:
  • Figure 3: $A_i$ can separate $S^2$ in many connected components.

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Theorem 4.4
  • Theorem 4.5
  • ...and 20 more