Dynamics of Newton's method for odd and even elliptic functions

TL;DR

The work addresses the dynamics of Newton's method for odd and even elliptic functions with a fixed lattice, establishing Julia set connectivity under absence of Herman rings and detailing the Fatou-set structure. It develops the framework via $\Lambda$-equivariance and torus projection, and proves that unbounded Fatou components, when they occur, are Baker or escaping domains, while all bounded components are simply connected. For triangular lattices, it proves the absence of Herman rings and shows that all Fatou components are bounded with connected Julia sets, and it demonstrates the existence of wandering domains coexisting with attracting basins in a parametric setting $N_b$ derived from $\wp_\Lambda+b$. These results extend the understanding of Newton maps beyond polynomials and entire functions, highlighting symmetry, lattice structure, and torus-quotient dynamics as key tools in elliptic function dynamics.

Abstract

We investigate Newton's method applied to any odd or any even elliptic function with an arbitrary period lattice. For any function of this type whose set of poles coincides with its period lattice, we show that the Julia set of its Newton map is connected, as long as no Herman rings exist. Moreover, we provide sufficient conditions on the Newton's method of any odd or even elliptic function to exhibit wandering domains coexisting with attracting basins. This phenomenon was first reported by Florido and Fagella for the Newton's method applied to a family of entire functions; however, our approach does not employ the lifting technique. We also provide a detailed study of a one-parameter family of elliptic functions given by $\wp_Λ+b$ with $Λ$ any triangular period lattice and $b\in\mathbb{C}$. We show that their associated Newton maps do not exhibit Herman rings or Baker domains, and any other Fatou component, including wandering, is bounded.

Figures (3)

• Figure 1: A sketch of the graphs $T_1$ and $T_2$ are shown in grey for the case $g_2=0$. Each $T_i$ connects a critical point $c_i$ with three poles of $N_b$, shown as blue dots, up to identification through a dashed side with its opposite side in $\mathcal{Q}$. The image of $T_1\cup T_2$ under $N_b$ is a curve $L_1\cup L_2$, symmetric with respect to $p$ and passing through $\infty$. Any curve $\alpha\in \Gamma$ has three preimages in $\mathcal{Q}$ up to identification, among them, $\alpha'$ is the only preimage surrounding $p$ and thus intersecting $T_1\cup T_2$ at the edge $[c_1, c_2]$.
• Figure 2: The dynamical plane for the Newton method of $\wp_\Lambda(z)+b$, where $\Lambda$ is triangular (in horizontal position) with invariants $g_2=0$ and $g_3\approx -12.8254$. As explained in Proposition \ref{['ex:wanderN_b']}, the parameter $b\approx -11.68$. The superattracting basins of the roots of $\wp_\Lambda(z)+b$ are shown in orange while two cycles of simply connected wandering domains are displayed in blue, with black and white arrows indicating their escaping orbits. The $\Lambda$-equivariance property implies the existence of infinitely many wandering domains with essentially two distinct orbital behaviours; one escaping to the left and one escaping to the right.
• Figure 3: (a) The $b$-parameter space for the Newton's method of $\wp_\Lambda(z)+b$ with $\Lambda$ triangular. Orange regions consist of parameters for which the free critical points of $N_b$ are trapped by the attracting basins of roots. Blue multibrots correspond to regions of parameters that give rise to wandering domains while light-blue multibrots represent parameters for which a free critical points belong to the basin of periodic cycles. (b) A magnified view of a "wandering" Multibrot inside the black rectangle in (a) is shown, with parameter $b\approx -11.68$ marked with a black dot.

Theorems & Definitions (43)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Remark 2.5
• Proposition 2.6