Unbounded components in parameter space of rational maps

Makienko, Peter

doi:10.1090/S1088-4173-00-00044-8

Unbounded components in parameter space of rational maps
HTML articles powered by AMS MathViewer

by Peter M. Makienko

Conform. Geom. Dyn. 4 (2000), 1-21

DOI: https://doi.org/10.1090/S1088-4173-00-00044-8

Published electronically: February 23, 2000

PDF | Request permission

Abstract:

Using pinching deformations of Riemann surfaces, we give several sufficient criteria for the space of quasiconformal deformations of rational map $R$ of degree $d$ to have non-compact closure in the space $Rat_{d}$ of rational maps of degree $d$ modulo conjugation by Möbius transformations.

References

Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367, DOI 10.24033/asens.1491
L. R. Goldberg and J. Milnor, Fixed points of polynomials maps, II. Fixed point portraits, Preprint Stony Brook (1990).
A. È. Erëmenko and G. M. Levin, Periodic points of polynomials, Ukrain. Mat. Zh. 41 (1989), no. 11, 1467–1471, 1581 (Russian); English transl., Ukrainian Math. J. 41 (1989), no. 11, 1258–1262 (1990). MR 1036486, DOI 10.1007/BF01056491
M. Yu. Lyubich, Dynamics of rational transformations: topological picture, Uspekhi Mat. Nauk 41 (1986), no. 4(250), 35–95, 239 (Russian). MR 863874
P. Makienko, Pinching and plumbing deformations of quadratic rational maps, Preprint, ICTP 93.32 (1993).
P. Makienko, Unbounded components in parameter space of rational maps, Preprint, ICTP (1993).
R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343, DOI 10.24033/asens.1446
Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386. MR 802500, DOI 10.5186/aasfm.1985.1042
Bernard Maskit, Parabolic elements in Kleinian groups, Ann. of Math. (2) 117 (1983), no. 3, 659–668. MR 701259, DOI 10.2307/2007038
Curt McMullen, Automorphisms of rational maps, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 31–60. MR 955807, DOI 10.1007/978-1-4613-9602-4_{3}
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
J. Milnor, Remarks on Quadratic maps, Preprint Stony Brook (1992).
K. Pilgrim, Ph.D. Thesis, 1994.
Mitsuhiro Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29. MR 892140, DOI 10.24033/asens.1522
Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, DOI 10.2307/1971308
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