On the Shapes of Rational Lemniscates

TL;DR

This work establishes that every lemniscate graph, including those with vertices, can be ε‑approximately realized as a rational lemniscate by a carefully constructed rational function with poles placed in prescribed grey-face locations. The authors reduce general graphs to the vertexless case via smoothing to analytic edges, and then build level-set approximations from sums of Green's functions, ultimately placing poles exactly through a fixed-point argument. They derive a quantitative Runge-type theorem with geometric convergence and prove an approximation of continua by Julia sets through rational maps with two attracting basins. The results unify topological, analytic, and dynamical perspectives on lemniscates, enabling control over topology, approximation rate, and Julia-set realizations with concrete geometric and harmonic-analytic tools.

Abstract

A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert's lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge's theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.

Figures (13)

• Figure 1: Pictured are three examples of lemniscate graphs, together with 2-colorings of their faces. For each example, the unbounded face is colored grey. The leftmost example consists of five disjoint Jordan curves and has no vertices. The center example has six connected components and six vertices (two of degree six and four of degree four). The right hand example has three connected components and one vertex of degree eight.
• Figure 2: We modify $\gamma$ near each vertex $v$ so it contains a line segment centered at $v$.
• Figure 3: We define a neighborhood $U$ of the upper half circle as a union of elliptical arcs, and define a homeomorphism of $U$ to itself by shifting the arcs. This is the identity outside $U$. When conjugated by $\phi$ this becomes an $\varepsilon$-homeomorphism of $G$ to $G'$ taking $e$ to $e'$, that is the identity on $G \setminus e$.
• Figure 4: Using linear fractional transformations and a square root, we can map the non-Jordan face of $\gamma$ to a Jordan domain. The inverse of this map is a rational map.
• Figure 5: If both endpoints of $e$ are the same, we can find a "figure 8" curve $\gamma$ containing $e$ and use a branch of $f^{-1}$ to map the non-Jordan face of $\gamma$ to a Jordan domain. Then the earlier construction gives an analytic approximation to $f(e)$ and applying the the rational map $f$ gives the an analytic approximation to $e$.

Theorems & Definitions (86)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem A
• Remark 1.5
• Corollary 1.6
• Theorem B
• Theorem C
• Theorem 2.1