A census of rational maps

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

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

MSC (2000):
Primary 37F10; Secondary 13P10

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

Published electronically:
April 4, 2000

MathSciNet review:
1749249

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

**1.**Davood Ahmadi,*Dynamics of certain rational maps of degree two*, Ph.D. thesis, University of Liverpool, 1991.**2.**G. Boccara,*Cycles comme produit de deux permutations de classes données*, Discrete Math.**38**(1982), no. 2-3, 129-142. MR**84e:20007****3.**Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks,*A census of cusped hyperbolic -manifolds*, Math. Comp.**68**(1999), no. 225, 321-332, With microfiche supplement. MR**99c:57035****4.**David Cox, John Little, and Donal O'Shea,*Ideals, varieties, and algorithms*, second ed., Springer-Verlag, New York, 1997, An introduction to computational algebraic geometry and commutative algebra. MR**97h:13024****5.**A. Douady and John Hubbard,*A proof of Thurston's topological characterization of rational functions*, Acta. Math.**171**(1993), no. 2, 263-297. MR**94j:58143****6.**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**85k:57005****7.**Adam L. Epstein,*Trace formulas for symmetric functions of the eigenvalues of a rational map*, in preparation.**8.**-,*Bounded hyperbolic components of quadratic rational maps*, SUNY Stony Brook Institute for Mathematical Sciences. Preprint (1997/99), no. 9, to appear,*Ergodic Theory Dynam. Systems*.**9.**Martin Hildebrand and Jeffrey Weeks,*A computer generated census of cusped hyperbolic -manifolds*, Computers and mathematics (Cambridge, MA, 1989), Springer, New York-Berlin, 1989, pp. 53-59. MR**90f:57043****10.**Curtis T. McMullen,*The classification of conformal dynamical systems*, Current developments in mathematics, 1995 (Cambridge, MA), Internat. Press, Cambridge, MA, 1994, pp. 323-360. MR**98h:58162****11.**John Milnor,*Hyperbolic components in spaces of polynomial maps*, SUNY Stonybrook Institute for Mathematical Sciences. Preprint (1992/93), no. 3.**12.**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**96b:58094****13.**Kevin M. Pilgrim,*Rational maps whose Fatou components are Jordan domains*, Ergodic Theory Dynam. Systems**16**(1996), no. 6, 1323-1343. MR**97k:58141****14.**Kevin M. Pilgrim,*Dessins d'enfants and Hubbard trees*, Ann. Sci. École Norm. Sup., to appear.**15.**-,*Cylinders for iterated rational maps*, Ph.D. thesis, University of California at Berkeley, May, 1994.**16.**Kevin M. Pilgrim and Tan Lei,*Combining rational maps and controlling obstructions*, Ergodic Theory Dynam. Systems**18**(1998), no. 1, 221-245. CMP**98:09****17.**Alfredo Poirier,*On post-critically finite polynomials part two: Hubbard trees*, SUNY Stony Brook Institute for Mathematical Sciences. Preprint (1993/97), no. 7.**18.**Mary Rees,*A partial description of the parameter space of rational maps of degree two: Part I*, Acta Math.**168**(1992), 11-87. MR**93f:58205****19.**W. P. Thurston,*Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds*, Ann. of Math.**124**(1986), 203-246. MR**88g:57014**

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Additional Information

**Eva Brezin**

Affiliation:
993 Amsterdam Ave., Apt. 4a, New York, NY 10025

Email:
ebrezin@bear.com

**Rosemary Byrne**

Affiliation:
Apt. 800, 1301 Massachusetts Ave. NW, Washington, DC 20005

Email:
rosemary.l.byrne@ccmail.census.gov

**Joshua Levy**

Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720

Email:
jdl@math.berkeley.edu

**Kevin Pilgrim**

Affiliation:
Department of Mathematics and Statistics, University of Missouri at Rolla, Rolla, MO 65409-0020

Email:
pilgrim@umr.edu

**Kelly Plummer**

Affiliation:
Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138

Email:
plummer@fas.harvard.edu

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

Keywords:
Complex dynamical systems,
Gr\"obner bases

Received by editor(s):
June 16, 1999

Received by editor(s) in revised form:
January 25, 2000

Published electronically:
April 4, 2000

Additional Notes:
Research supported in part by the National Science Foundation’s Research Experiences for Undergraduates program.

The fourth author’s research was partially supported by the NSF’s REU program at Cornell, and by NSF Grants DMS-9703724 and DMS-9996070

Article copyright:
© Copyright 2000
American Mathematical Society