A census of rational maps
Authors:
Eva Brezin, Rosemary Byrne, Joshua Levy, Kevin Pilgrim and Kelly Plummer
Journal:
Conform. Geom. Dyn. 4 (2000), 3574
MSC (2000):
Primary 37F10; Secondary 13P10
Published electronically:
April 4, 2000
MathSciNet review:
1749249
Fulltext PDF Free Access
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Additional Information
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
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 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. 23, 129142. 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, 321332, With microfiche supplement. MR 99c:57035
 4.
 David Cox, John Little, and Donal O'Shea, Ideals, varieties, and algorithms, second ed., SpringerVerlag, 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, 263297. 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, 773790. 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 YorkBerlin, 1989, pp. 5359. 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. 323360. MR 98h:58162
 11.
 John Milnor, Hyperbolic components in spaces of polynomial maps, SUNY Stonybrook Institute for Mathematical Sciences. Preprint (1992/93), no. 3.
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 John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 3783. 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, 13231343. 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, 221245. CMP 98:09
 17.
 Alfredo Poirier, On postcritically 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), 1187. MR 93f:58205
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 W. P. Thurston, Hyperbolic structures on 3manifolds I: Deformation of acylindrical manifolds, Ann. of Math. 124 (1986), 203246. 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 654090020
Email:
pilgrim@umr.edu
Kelly Plummer
Affiliation:
Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138
Email:
plummer@fas.harvard.edu
DOI:
http://dx.doi.org/10.1090/S1088417300000503
PII:
S 10884173(00)000503
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 DMS9703724 and DMS9996070
Article copyright:
© Copyright 2000 American Mathematical Society
