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A universal model for the bifurcations of asymptotic values

Astorg Matthieu, Benini Anna Miriam, Fagella Núria

Abstract

We study the notion of tangent-like maps, which is a transcendental analogue of polynomial-like maps. We introduce a model family analogous to quadratic polynomials, with only one free asymptotic value, and define the "Tandelbrot set" as the analogue of the Mandelbrot set. We prove a Straightening Theorem for tangent-like maps, with uniqueness of the model map in the case where the filled-in Julia set is connected, and a parameter version of the Straightening Theorem for suitable holomorphic families of tangent-like maps. As a consequence, we prove the existence of topological copies of the Tandelbrot set in bifurcation loci of numerous families of meromorphic maps.

A universal model for the bifurcations of asymptotic values

Abstract

We study the notion of tangent-like maps, which is a transcendental analogue of polynomial-like maps. We introduce a model family analogous to quadratic polynomials, with only one free asymptotic value, and define the "Tandelbrot set" as the analogue of the Mandelbrot set. We prove a Straightening Theorem for tangent-like maps, with uniqueness of the model map in the case where the filled-in Julia set is connected, and a parameter version of the Straightening Theorem for suitable holomorphic families of tangent-like maps. As a consequence, we prove the existence of topological copies of the Tandelbrot set in bifurcation loci of numerous families of meromorphic maps.
Paper Structure (25 sections, 71 theorems, 144 equations, 10 figures)

This paper contains 25 sections, 71 theorems, 144 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Holomorphic motions, quasiconformal maps
  4. Polynomial-like maps
  5. Meromorphic maps, families and bifurcations
  6. Tracts over asymptotic values
  7. Tangent-like maps
  8. First definitions: tangent-like maps and hybrid equivalence
  9. Connectedness of $K(f)$
  10. The model family
  11. First properties
  12. Conformal conjugacy on the attracting basin of $0$
  13. Rigidity of conjugacies near the Julia set
  14. Parameter plane: The Tandelbrot set. Proof of Theorem B
  15. Conjugacy classes in $\mathcal{T}$
  16. ...and 10 more sections

Key Result

Corollary 1.2

Let $\{f_\lambda\}_{\lambda \in \Lambda}$ be a natural family of finite type meromorphic maps. Assume that the following conditions hold: Then, near any $\lambda_1 \in \mathrm{Bif}$, there exists a topological embedding of $\partial \mathcal{M}_d$ (for some $d \geq 2$) or $\partial \mathcal{T}$ contained in $\mathrm{Bif}$.

Figures (10)

  • Figure 1: The definition of a tangent-like map. The point $u \in \partial U$ is an essential singularity. The asymptotic value $v$ can be located anywhere in $V$.
  • Figure 2: Left: The filled Julia set $K(T_\alpha)$ of $T_\alpha$ for $\alpha\simeq-0.021+i 0.009$, showing the basin of attraction of an attracting 3-periodic orbit. The asymptotic value is $v=1/\alpha$. Right: A quasiconformal copy of $K(T_\alpha)$ in the dynamical plane of a Newton's method $N_a$, applied to the family of entire functions $f_a(z)=z+ a e^z$ for $a\simeq-1.1627+i 0.1143$. There is a 3-periodic orbit under $N_a^2$. The point $u$ is a pole of $N_a$ which becomes an essential singularity under the second iterate. The assymptotic value is $v=0$.
  • Figure 3: In black, the Tandelbrot set.
  • Figure 4: Sketch of the tangent-like renormalization scheme. Suppose $\lambda_0$ is a virtual cycle parameter for which $f_{\lambda_0}^n(v_{\lambda_0})=\infty$. Then, for nearby parameters $\lambda$ the domain $U_\lambda$ of the tangent-like restriction is a tract above a disk $D_\lambda$ containing the asymptotic value $v_\lambda$. The iterate $f_\lambda^{n+1}$ maps $D_\lambda$ to another disk containing the pole $p_\lambda$, which is then mapped to a fixed disk $V$ centered at infinity. Hence, the map $f_\lambda^{n+3}: U_\lambda \to V \setminus \{f_\lambda^{n+2}(v_\lambda)\}$ is tangent-like.
  • Figure 5: Left: A copy of $\mathcal{T}$ in the parameter space of the Newton's method of $f_a(z)=z+ae^z$, illustrating Theorem C. Right: A small copy of $\mathcal{T}$ inside $\mathcal{T}$, illustrating Theorem D.
  • ...and 5 more figures

Theorems & Definitions (163)

  • Definition 1.1: Multibrot sets
  • Corollary 1.2
  • Corollary 1.3: Self-similarity of $\mathcal{T}$
  • Definition 2.1: Holomorphic motion
  • Proposition 2.2: Mañé-Sad-Sullivan's $\lambda$-lemma, mss
  • Theorem 2.3: Slodkowski's extension slo
  • Lemma 2.4: Boundary extension of conformal maps of $\mathbb{D}$ pom
  • Lemma 2.5: Interpolation in quasi-annuli EFGP
  • Definition 2.6: Polynomial-like map
  • Definition 2.7: Filled Julia set
  • ...and 153 more