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Meromorphic Functions with a Polar Asymptotic Value

Tao Chen, Linda Keen

Abstract

This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, $\exp(z) + a$, and the tangent family $λ\tan z$. The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values $\pm λi$, is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between $\exp(z) + a$ and $λ\tan z$. Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be pole, the "polar asymptotic value" of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the $\exp(z) + a$ and $λ\tan z$ families.

Meromorphic Functions with a Polar Asymptotic Value

Abstract

This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, , and the tangent family . The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values , is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between and . Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be pole, the "polar asymptotic value" of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the and families.
Paper Structure (23 sections, 36 theorems, 71 equations, 10 figures)

This paper contains 23 sections, 36 theorems, 71 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Dynamics
  3. The family ${\mathcal{FP}}_2$
  4. Combinatorics of the prepoles
  5. L-R Structure of the pre-asymptotic tracts in the dynamic plane
  6. Virtual cycles
  7. Virtual cycles in $\mathcal{FP}_2$
  8. Virtual multipliers
  9. Pseudo-hyperbolic maps
  10. Kneading sequences
  11. Examples
  12. The Parameter plane of $\mathcal{FP}_2$
  13. Pseudo-hyperbolic components
  14. The set $\mathcal{H}_1$
  15. Properties of $\Omega_n$
  16. ...and 8 more sections

Key Result

Theorem 1

Let $f \in {\mathcal{FP}}_2$ and suppose the free asymptotic value is also a prepole. Then the Julia set of $f$ contains Cantor bouquets at infinity as well as at each of the poles and prepoles; if, in addition, $f$ has an attracting fixed point, the stable set is a completely invariant connected se

Figures (10)

  • Figure 1: The first two pullbacks of the left and right halfplanes in the L-R structures of $f_{\lambda}$ for $\lambda=-3+i\pi/2$ on the left and $\lambda=-3+i\pi/2$ on the right.
  • Figure 2: Three pullbacks of the left and right half plane in L-R structure of $f_{\lambda}$ for $\lambda=-3+i\pi/2$.
  • Figure 3: Virtual cycle period $2$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
  • Figure 4: Virtual cycle period $4$
  • Figure 6: A period $5$ example that shows the kneading sequence cannot be $*0000$.
  • ...and 5 more figures

Theorems & Definitions (80)

  • Theorem : Theorem A
  • Theorem : Theorem B
  • Theorem : Theorem C
  • Theorem : Theorem D
  • Definition 1
  • Remark 3.1
  • Definition 2
  • Definition 3
  • Proposition 3.1
  • proof
  • ...and 70 more