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On the Combinatorial Rigidity for Polynomials with Attracting Cycles

Yueyang Wang

Abstract

We show that every polynomial of degree $d \geq 2$ in the connectedness locus with an attracting cycle which attracts at least two critical points and no indifferent cycles is not combinatorially rigid. In particular, we prove that a hyperbolic polynomial with connected Julia set is combinatorially rigid if and only if it is of the ``disjoint type''.

On the Combinatorial Rigidity for Polynomials with Attracting Cycles

Abstract

We show that every polynomial of degree in the connectedness locus with an attracting cycle which attracts at least two critical points and no indifferent cycles is not combinatorially rigid. In particular, we prove that a hyperbolic polynomial with connected Julia set is combinatorially rigid if and only if it is of the ``disjoint type''.
Paper Structure (18 sections, 15 theorems, 32 equations, 6 figures)

This paper contains 18 sections, 15 theorems, 32 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main Results
  4. Idea of the proof and structure of the paper
  5. Preliminary
  6. Notations
  7. External rays
  8. Polynomial-like maps
  9. Perturbation of the Free Critical Orbits
  10. Unique Fatou critical point with infinite orbit
  11. Space with marked critical points
  12. Sectors, quadrilaterals, limbs
  13. Change the dynamical position
  14. Dynamic and Lamination of the Limit Maps
  15. The limit map
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $d\geq 3$, every polynomial $f \in \mathcal{C}_d$ with no indifferent cycle and possessing an attracting cycle which attracts at least two critical points counted with multiplicity is not combinatorially rigid.

Figures (6)

  • Figure 1: $K(f_0)$
  • Figure 2: $K(f)$
  • Figure 3: $S_{{\bf a}}(\mathcal{I})$
  • Figure 4: $Q_{{\bf a}}(\mathcal{I},s,s')$
  • Figure 5: $K(f)$
  • ...and 1 more figures

Theorems & Definitions (43)

  • Definition 1.1: rational lamination
  • Definition 1.2: combinatorial rigidity
  • Theorem 1.1: main result
  • Theorem 1.2: hyperbolic case
  • Example 1.1
  • Lemma 2.1: holomorphic motion of external rays
  • Lemma 2.2: continuity of external rays landing on repelling orbit
  • Theorem 2.3: Roesch-Yin roesch2008boundary
  • Proposition 2.4: straightening theorem
  • proof
  • ...and 33 more