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Christopher J. Bishop

Publications and source records attributed to Christopher J. Bishop.

At least 19 recordsLinked to original sources
arXiv2025

Non-compact Riemann surfaces are equilaterally triangulable

We show that every open Riemann surface can be obtained by glueing together a countable collection of equilateral triangles, in such a way that every vertex belongs to finitely many triangles. Equivalently, it is a _Belyi surface_: There exists a holomorphic branched covering to the Riemann sphere that is branched only over three values. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.

math.CV↗
arXiv2025

On the Shapes of Rational Lemniscates

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.

math.CV↗
arXiv2025

Models for the Eremenko-Lyubich class

If $f$ is in the Eremenko-Lyubich class (transcendental entire functions with bounded singular set) then $Ω= \{ z: |f(z)| > R\}$ and $f|_Ω$ must satisfy certain simple topological conditions when $R$ is sufficiently large. A model $(Ω, F)$ is an open set $Ω$ and a holomorphic function $F$ on $Ω$ that satisfy these same conditions. We show that any model can be approximated by an Eremenko-Lyubich function in a precise sense. In many cases, this allows the construction of functions in the Eremenko-Lyubich with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do.

math.CV↗
arXiv2025

Models for the Speiser class

The Eremenko-Lyubich class consists of transcendental entire functions with bounded singular set and the Speiser class is made up of functions with a finite singular set. In an earlier paper "Models for the Eremenko-Lyubich class" I gave a method for constructing Eremenko-Lyubich functions that approximate certain simpler functions called models. In this paper, I show that all such models can be approximated in a weaker sense by Speiser class functions, and that the stronger approximation possible using Eemenko-Lyubich functions can fail for the Speiser class. In particular, I give geometric restrictions on the geometry of a Speiser class function that need not be satisfied by general Eremenko-Lyubich functions.

math.CV↗
arXiv2023

A Geometric Approach to Polynomial and Rational Approximation

We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.

math.CV↗
arXiv2022

Equilateral Triangulations and The Postcritical Dynamics of Meromorphic Functions

We show that any dynamics on any planar set $S$ discrete in some domain $D$ can be realized by the postcritical dynamics of a function holomorphic in $D$, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain $D$ can be equilaterally triangulated with triangles whose diameters $\rightarrow0$ (at any prescribed rate) near $\partial D$.

math.DS↗
arXiv2020

Quadrilateral meshes for PSLGs

We prove that every planar straight line graph with $n$ vertices has a conforming quadrilateral mesh with $O(n^2)$ elements, all angles $\leq 120^\circ$ and all new angles $\geq 60^\circ$. Both the complexity and the angle bounds are sharp. Moreover, all but $O(n)$ of the angles may be taken in a smaller interval, say $[89^\circ, 91^\circ]$.

cs.CG↗
arXiv2020

Nonobtuse triangulations of PSLGs

We show that any planar straight line graph (PSLG) with $n$ vertices has a conforming triangulation by $O(n^{2.5})$ nonobtuse triangles (all angles $\leq 90^\circ$), answering the question of whether any polynomial bound exists. A nonobtuse triangulation is Delaunay, so this result also improves a previous $O(n^3)$ bound of Eldesbrunner and Tan for conforming Delaunay triangulations of PSLGs. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only $O(n^2)$ triangles are needed, improving an $O(n^4)$ bound of Bern and Eppstein. We also show that for any $ε>0$, every PSLG has a conforming triangulation with $O(n^2/ε^2)$ elements and with all angles bounded above by $90^\circ + ε$. This improves a result of S. Mitchell when $ε= 3 π/8 = 67.5^\circ $ and Tan when $ε= 7π/30 =42^\circ$.

cs.CG↗
arXiv2020

Speiser class Julia sets with dimension near one

For any $ δ>0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=δ$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.

math.CV↗
arXiv2020

Conformal mapping in linear time

Given any $ε>0$ and any planar region $Ω$ bounded by a simple n-gon $P$ we construct a ($1 + ε)$-quasiconformal map between $Ω$ and the unit disk in time $C(ε)n$. One can take $ C(ε) = C + C \log (1/ε) \log \log (1/ε)$.

math.CV↗
arXiv2020

The order conjecture fails in S

We construct an entire function $f$ with only three singular values whose order of growth can change under a quasiconformal equivalence. This is a counterexample to the Order Conjecture in the Speiser class ${\mathcal S}$ of entire functions.

math.CV↗
arXiv2020

True trees are dense

We show that any compact, connected set $K$ in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, $K$ can be approximated in the Hausdorff metric by a true tree in the sense of Grothendieck's dessins d'enfants.

math.CV↗
arXiv2019

Prescribing the Postsingular Dynamics of Meromorphic Functions

We show that any dynamics on any discrete planar sequence $S$ can be realized by the postsingular dynamics of some transcendental meromorphic function, provided we allow for small perturbations of $S$. This work was influenced by an analogous result of DeMarco, Koch and McMullen for finite $S$ in the rational setting. The proof contains a method for constructing meromorphic functions with good control over both the postsingular set of $f$ and the geometry of $f$, using the Folding Theorem of Bishop and a classical fixpoint theorem of Tychonoff.

math.CV↗
arXiv2018

Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler Geometry

It is known that the almost-Kaehler anti-self-dual metrics on a given 4-manifold sweep out an open subset in the moduli space of anti-self-dual metrics. However, we show here by example that this subset is not generally closed, and so need not sweep out entire connected components in the moduli space. Our construction hinges on an unexpected link between harmonic functions on certain hyperbolic 3-manifolds and self-dual harmonic 2-forms on associated 4-manifolds.

math.DG↗
arXiv2016

Qusisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if $f:X\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then $\dim_H f(E)\leq\dim_H E$ for "almost every" bounded Ahlfors regular set $E\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then $\dim_H f(E)=\dim_H E$ for "almost every" Ahlfors regular set $E\subset X$. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if $f$ is a quasiconformal map of $\mathbb{R}^N$, $N\geq 2$, then for Lebesgue a.e. $y\in\mathbb{R}^N$ we have $\dim_H f(y+E) = \dim_H E$. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if $E \subset \mathbb{R}$ is Ahlfors $d$-regular, $d<1$, then some component of $f(E \times \mathbb{R})$ has dimension at most $2/(d+1)$, and we construct examples to show this bound is sharp. In addition, we show there is a $1$-dimensional set $S\subseteq \mathbb R$ and planar quasiconformal map $f$ such that $f(\mathbb{R} \times S)$ contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson \cite{Tyson:frequency} and answer questions posed in \cite{Tyson:frequency} and \cite{AimPL}.

math.CV↗
arXiv1998

Non-removable sets for quasiconformal and locally biLipschitz mappings in R^3

We give an example of a totally disconnected set E in R^3 which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism f of R^3 to itself which is quasiconformal off E, but not quasiconformal on all of R^3. The set E may be taken with Hausdorff dimension 2. The construction also gives a non-removable set for locally biLipschitz homeomorphisms.

math.CV↗