Some rational maps whose Julia sets are not locally connected

Roesch, P.

doi:10.1090/S1088-4173-06-00139-1

Some rational maps whose Julia sets are not locally connected
HTML articles powered by AMS MathViewer

by P. Roesch

Conform. Geom. Dyn. 10 (2006), 125-135

DOI: https://doi.org/10.1090/S1088-4173-06-00139-1

Published electronically: July 6, 2006

PDF | Request permission

Abstract:

We describe examples of rational maps which are not topologically conjugate to a polynomial and whose Julia sets are connected but not locally connected.

References

James W. Anderson and Bernard Maskit, On the local connectivity of limit set of Kleinian groups, Complex Variables Theory Appl. 31 (1996), no. 2, 177–183. MR 1423249, DOI 10.1080/17476939608814957

A. Douady, J. H. Hubbard

Etude dynamique des polynômes complexes

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
Adrien Douady, Disques de Siegel et anneaux de Herman, Astérisque 152-153 (1987), 4, 151–172 (1988) (French). Séminaire Bourbaki, Vol. 1986/87. MR 936853
Étienne Ghys, Transformations holomorphes au voisinage d’une courbe de Jordan, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 16, 385–388 (French, with English summary). MR 748928

M. Herman

Conjugaison quasi-symétrique des difféomorphismes du cercle à desrotations et applications aux disques singuliers de Siegel, I.

John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
John Milnor, Local connectivity of Julia sets: expository lectures, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 67–116. MR 1765085
Yair N. Minsky, On rigidity, limit sets, and end invariants of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 7 (1994), no. 3, 539–588. MR 1257060, DOI 10.1090/S0894-0347-1994-1257060-3
Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
Curtis T. McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001), no. 1, 35–91. MR 1859018, DOI 10.1007/PL00005809

P. Roesch

On local connectivity for the Julia set of rational maps

Dan Erik Krarup Sørensen, Describing quadratic Cremer point polynomials by parabolic perturbations, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 739–758. MR 1631760, DOI 10.1017/S0143385798108301
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

M. Shishikura

The connectivity of the Julia set of rational maps and Fixed points

Lei Tan and Yongcheng Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A 39 (1996), no. 1, 39–47. MR 1397233
Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095, DOI 10.1090/coll/028

S. Zakeri

In Shahyad, a volume dedicated to Siavash Shahshahani on the occasion of his 60th birthday, 2002.