On the Shapes of Rational Lemniscates

TL;DR

This paper demonstrates that any planar Euler graph can be closely approximated by a homeomorphic rational lemniscate, extending classical approximation theorems and connecting rational lemniscates with Julia sets.

Contribution

It generalizes Hilbert's lemniscate theorem to rational lemniscates and establishes new approximation results for planar continua using Julia sets.

Findings

Any planar Euler graph can be approximated by a homeomorphic rational lemniscate.

Provides a sharp quantitative version of Runge's theorem on rational approximation.

Shows planar continua can be approximated by Julia sets of rational maps.

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.