A census of rational maps

Brezin, Eva; Byrne, Rosemary; Levy, Joshua; Pilgrim, Kevin; Plummer, Kelly

doi:10.1090/S1088-4173-00-00050-3

A census of rational maps
HTML articles powered by AMS MathViewer

by Eva Brezin, Rosemary Byrne, Joshua Levy, Kevin Pilgrim and Kelly Plummer

Conform. Geom. Dyn. 4 (2000), 35-74

DOI: https://doi.org/10.1090/S1088-4173-00-00050-3

Published electronically: April 4, 2000

PDF | Request permission

Abstract:

We discuss the general combinatorial, topological, algebraic, and dynamical issues underlying the enumeration of postcritically finite rational functions, regarded as holomorphic dynamical systems on the Riemann sphere. We present findings from our creation of a census of all degree two and three hyperbolic nonpolynomial maps with four or fewer postcritical points. Our data is tabulated in detail at http://www.umr.edu/~pilgrim/Research/Census/WebPages/Main/Main.html

References

Dynamics of certain rational maps of degree two

G. Boccara, Cycles comme produit de deux permutations de classes données, Discrete Math. 38 (1982), no. 2-3, 129–142 (French). MR 676530, DOI 10.1016/0012-365X(82)90282-5
Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks, A census of cusped hyperbolic $3$-manifolds, Math. Comp. 68 (1999), no. 225, 321–332. With microfiche supplement. MR 1620219, DOI 10.1090/S0025-5718-99-01036-4
David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. An introduction to computational algebraic geometry and commutative algebra. MR 1417938
Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
Allan L. Edmonds, Ravi S. Kulkarni, and Robert E. Stong, Realizability of branched coverings of surfaces, Trans. Amer. Math. Soc. 282 (1984), no. 2, 773–790. MR 732119, DOI 10.1090/S0002-9947-1984-0732119-5

Trace formulas for symmetric functions of the eigenvalues of a rational map

Bounded hyperbolic components of quadratic rational maps

Ergodic Theory Dynam. Systems

Martin Hildebrand and Jeffrey Weeks, A computer generated census of cusped hyperbolic $3$-manifolds, Computers and mathematics (Cambridge, MA, 1989) Springer, New York, 1989, pp. 53–59. MR 1005959
Curtis T. McMullen, The classification of conformal dynamical systems, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 323–360. MR 1474980

Hyperbolic components in spaces of polynomial maps

John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 37–83. With an appendix by the author and Lei Tan. MR 1246482
Kevin M. Pilgrim, Rational maps whose Fatou components are Jordan domains, Ergodic Theory Dynam. Systems 16 (1996), no. 6, 1323–1343. MR 1424402, DOI 10.1017/S0143385700010051

Dessins d’enfants and Hubbard trees

Cylinders for iterated rational maps

Combining rational maps and controlling obstructions

18

On post-critically finite polynomials part two: Hubbard trees

Mary Rees, A partial description of parameter space of rational maps of degree two. I, Acta Math. 168 (1992), no. 1-2, 11–87. MR 1149864, DOI 10.1007/BF02392976
William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI 10.2307/1971277