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Hyperbolic components of cosine family with a fixed critical point

Weiyuan Qiu, Lingrui Wang

Abstract

We studied the parameter plane of the cosine functions with a fixed critical point. The hyperbolic components can be classified into three types: A, C and D. All the hyperbolic components are bounded and simply connected, except for the unique type-A component, which contains 0 as an isolated boundary point. Using the method of para-puzzle, we constructed a phase-parameter transfer mapping and proved that the boundaries of hyperbolic components are Jordan curves. By a similar idea, the hyperbolic components of type C are quasidisks.

Hyperbolic components of cosine family with a fixed critical point

Abstract

We studied the parameter plane of the cosine functions with a fixed critical point. The hyperbolic components can be classified into three types: A, C and D. All the hyperbolic components are bounded and simply connected, except for the unique type-A component, which contains 0 as an isolated boundary point. Using the method of para-puzzle, we constructed a phase-parameter transfer mapping and proved that the boundaries of hyperbolic components are Jordan curves. By a similar idea, the hyperbolic components of type C are quasidisks.
Paper Structure (20 sections, 56 theorems, 31 equations, 1 figure)

This paper contains 20 sections, 56 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic Properties
  3. Parameterization of hyperbolic components
  4. Hyperbolic components of type A
  5. Hyperbolic components of type C
  6. Hyperbolic components of type D
  7. Parameter rays
  8. Dynamic rays
  9. Parameter rays
  10. Parameter internal rays
  11. Para-puzzles
  12. Maps on the boundaries of hyperbolic components
  13. Construction of para-puzzles
  14. Holomorphic motion of para-puzzle pieces
  15. transfer from the dynamical plane to the parameter plane
  16. ...and 5 more sections

Key Result

Theorem 1.1

Every hyperbolic component is a bounded domain in $\mathbb{C}$. The hyperbolic component of type A is unique, which is homeomorphic to the punctured unit disk. Hyperbolic components of type C and D are all homeomorphic to the unit disk.

Figures (1)

  • Figure 1: Parameter space of $f_v$, black regions are hyperbolic components.

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • Corollary 2.2
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 88 more