William J. Floyd
Research Subjects
William J. Floyd has a total of 14 co-authors. Here are the top co-authors by number of shared publications:
- Walter Parry
Eastern Michigan University
30 shared publications - James W. Cannon
Department of Mathematics, Brigham Young University, Provo, Utah 84602
16 shared publications - Kevin M. Pilgrim
Indiana University
14 shared publications - Edgar Saenz
Virginia Tech
3 shared publications J. W. Cannon
3 shared publications- Sarah Koch
University of Michigan
3 shared publications
Latest Publications
- Open Access
Realizing polynomial portraits
It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of $f$. We encode the orbits of the critical points of $f$ with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial portrait is realized as the ramification portrait of a postcritically finite polynomial, and classify which abstract polynomial portraits can only be realized by unobstructed maps.Open AccessRealizing polynomial portraits
It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of $f$. We encode the orbits of the critical points of $f$ with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial portrait is realized as the ramification portrait of a postcritically finite polynomial, and classify which abstract polynomial portraits can only be realized by unobstructed maps. - 28 March 2022
- Published by Springer Nature
- p. 401-420
The Pullback Map on Teichmüller Space Induced from a Thurston Map
- 28 March 2022
- Published by Springer Nature
- p. 401-420
The Pullback Map on Teichmüller Space Induced from a Thurston Map
- 16 January 2021
- Published by Springer Nature in Geometriae Dedicata
- Vol. 213 (1) , 487-512
Rationality is decidable for nearly Euclidean Thurston maps
- 16 January 2021
- Published by Springer Nature in Geometriae Dedicata
- Vol. 213 (1) , 487-512
Rationality is decidable for nearly Euclidean Thurston maps
- 14 January 2020
- Published by American Mathematical Society (AMS) in Conformal Geometry and Dynamics of the American Mathematical Society
- Vol. 24 (2) , 29-50
Expansion properties for finite subdivision rules II
We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.
- 14 January 2020
- Published by American Mathematical Society (AMS) in Conformal Geometry and Dynamics of the American Mathematical Society
- Vol. 24 (2) , 29-50
Expansion properties for finite subdivision rules II
We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.
- Open Access
Expansion properties for finite subdivision rules II
We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.
Open AccessExpansion properties for finite subdivision rules II
We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.
- 1 January 2019
- Published by Institute of Mathematics, Polish Academy of Sciences in Fundamenta Mathematicae
- Vol. 244 (1) , 49-72
Presentations of NET maps
A branched covering $f\colon S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. We show that up to equivalence, each NET map admits a normal form in terms of si- 1 January 2019
- Published by Institute of Mathematics, Polish Academy of Sciences in Fundamenta Mathematicae
- Vol. 244 (1) , 49-72
Presentations of NET maps
A branched covering $f\colon S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. We show that up to equivalence, each NET map admits a normal form in terms of si- Open Access
Rationality is decidable for nearly Euclidean Thurston maps
Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound $C$, there are finitely many diagrams of size at most $C$. Given a NET map $F$ presented by a diagram of size at most $C$, the problem of determining whether $F$ is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation in terms of $C$ and one other natural geometric quantity. This result partially explains the observed effectiveness of the computer program NETmap in deciding rationality.Open AccessRationality is decidable for nearly Euclidean Thurston maps
Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound $C$, there are finitely many diagrams of size at most $C$. Given a NET map $F$ presented by a diagram of size at most $C$, the problem of determining whether $F$ is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation in terms of $C$ and one other natural geometric quantity. This result partially explains the observed effectiveness of the computer program NETmap in deciding rationality. - 22 November 2018
- Published by Springer Nature in Science China Mathematics
- Vol. 61 (12) , 2237-2266
Expansion properties for finite subdivision rules I
- 22 November 2018
- Published by Springer Nature in Science China Mathematics
- Vol. 61 (12) , 2237-2266
Expansion properties for finite subdivision rules I
- 22 August 2017
- Published by Association for Mathematical Research in Arnold Mathematical Journal
- Vol. 3 (3) , 365-395
Origami, Affine Maps, and Complex Dynamics
We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.
- 22 August 2017
- Published by Association for Mathematical Research in Arnold Mathematical Journal
- Vol. 3 (3) , 365-395
Origami, Affine Maps, and Complex Dynamics
We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.
- Open Access
Modular groups, Hurwitz classes and dynamic portraits of NET maps
An orientation-preserving branched covering $f: S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. Inspired by classical, non-dynamical notions such as Hurwitz equivalence of branched covers of surfaces, we develop invariants for such maps. We then apply these notions to the classification and enumeration of NET maps. As an application, we obtain a complete classification of the dynamic critical orbit portraits of NET maps.Open AccessModular groups, Hurwitz classes and dynamic portraits of NET maps
An orientation-preserving branched covering $f: S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. Inspired by classical, non-dynamical notions such as Hurwitz equivalence of branched covers of surfaces, we develop invariants for such maps. We then apply these notions to the classification and enumeration of NET maps. As an application, we obtain a complete classification of the dynamic critical orbit portraits of NET maps.