Paper Fortune Tellers in the combinatorial dynamics of some generalized McMullen maps with both critical orbits bounded

TL;DR

This work studies the generalized McMullen maps $F_{n,a,b}(z)=z^n+a/z^n+b$ with $n\ge3$ and $a\neq0$, revealing a novel dynamical regime when both free critical orbits are bounded: the Julia set $J(F_{n,a,b})$ can contain infinitely many copies of a quadratic Julia set and, simultaneously, infinitely many subsets that are alternately identified by ray-pairings to form an altered quadratic Julia set. The authors develop a combinatorial construction that transfers external angle data from a quadratic baby Julia set to its $n$-fold symmetric copies, yielding a tree of preimages $J_*$ with angle assignments that reflect the dynamics via angle-doubling on preimages containing a critical point and identity elsewhere. By focusing on the basilica case, they classify how the placement of the second critical value $v_-$ within a preimage copy of the baby set alters the lamination, producing a taxonomy of altered baby Julia sets (Types 0, 1-1, 1-2, 2, N) and a general inductive mechanism for arbitrarily many steps, thereby demonstrating a rich combinatorial mechanism for producing non-quadratic-like Julia subsets inside $J(F_{n,a,b})$. The results illuminate how local critical-point structure can globally reshape the Julia set’s combinatorial model and open avenues for extending such phenomena to other quadratic baby Julia sets. The work has implications for understanding the diversity of Julia set topologies within parameter spaces of rational maps and for the broader study of ray-identification dynamics in complex dynamics.

Abstract

For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for, some new dynamical behavior. In particular, we describe a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets and infinitely many subsets homeomorphic to a set which is obtained by starting with a quadratic Julia set, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we will describe.

Figures (17)

• Figure 1: Portions of the Julia set of $F_{n,a,b}$ for $n=5$, $a=0.1317-0.0073i$, $b=0.03+0.02i$. The right image is homeomorphic to a quadratic Julia set, it is a preimage copy under $F_{n,a,b}$ of a baby quadratic Julia set in $J(F_{n,a,b})$ associated with the critical value $v_+$. The left is a preimage of the right under $F_{n,a,b}$, and is what we refer to as "altered". The red dot in the center of the right image marks the location of the other critical value $v_-$; note this is not in the location of a critical value in a basilica Julia set, which would be in one of the largest Fatou components adjacent to the central component. This alternate placement is what causes the altered shape of the preimage.
• Figure 2: The Julia set for the basilica $P_{-1}(z)=z^2-1$, shown with selected external rays and letters we will use to refer to the larger Fatou components.
• Figure 3: Lamination diagram for the basilica $P_{-1}(z)$
• Figure 4: The Julia set of $F_{n,a,b}$ for $n=3$, $a=0.05855-0.01282i$, $b=0.02+0.03i$. The baby quadratic Julia set is on the positive real axis. Its $n-1=2$ rotationally symmetric preimages are apparent, as are several smaller deeper-level preimages. The altered preimage is several preimages deep, so it is too small to see in detail without zooming in to the area outlined by the red square.
• Figure 5: Topological conjugacy diagram for Theorem \ref{['thm:angle assignments']}. On $J_+$, the map $F$ is conjugate to a $P_c$ on its Julia set $J_c$, and hence the angle assignments $\gamma_c$ for $J_c$ can be passed to $J_+$.

Theorems & Definitions (22)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• Corollary 3.3