Continuity of Julia sets and invariant rays

Abstract

For certain typical perturbations $(f_n)_n$ of a rational map $f$ with parabolic cycles, we investigate the relations between the Hausdorff convergence of Julia sets and invariant rays, and the horocyclic convergence of multipliers of periodic points. We establish several equivalent characterizations by means of parabolic implosion theory. This builds upon an analysis of the edge dynamics on the tree for the gate structure induced by the perturbation. The edge dynamics which are driven by Oudkerk's algorithm, are used to trace the orbits for the near parabolic perturbations.

Figures (15)

• Figure 1: The Double Mandelbrot set $\{\lambda\in \mathbb C; J(\lambda z+z^2) \text{ is connected}\}$ (up), the Julia sets $J(\lambda z+z^2)$ for $\lambda=1-\varepsilon$ and $\lambda=1+\varepsilon i$ with small $\varepsilon>0$. According to Theorem \ref{['appe']}, $J(\lambda_n z+z^2)\rightarrow J(z+z^2) \Longleftrightarrow |{\rm Re}(1/(1-\lambda_n))|\rightarrow +\infty$.
• Figure 2: Julia sets of $f_{a,b}(z)=z+z((z-a)^2+b)$, where $(a,b)=(0,0)$ (left) and $(0.2i, -0.0005539+0.00012975i)$ (right). This example is provided by Jie Cao, and shows the case that $f_{a,b}\rightarrow f_{0,0}$$1$-horocyclically at $0$.
• Figure 3: The sets $U_{k, s, f_0, \phi}$, in the case $\nu=3$.
• Figure 4: Two cases of well behaved map $f$. In both cases, $\nu=3$ and $\sigma_0(f), \sigma_1(f), \sigma_2(f)$ are fixed points of $f$. This Figure comes from Oudkerk's thesis Ou99. The gate structure $\mathbf G=(2,1,*)$ for (a), and $\mathbf G=(1,*,3)$ for (b).
• Figure 5: An example of gate structure $\mathbf G=(2,3,1,4,5)$, in this case $\nu=5$.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 2.1
• Proposition 2.2