## Some rational maps whose Julia sets are not locally connected

HTML articles powered by AMS MathViewer

- by P. Roesch PDF
- Conform. Geom. Dyn.
**10**(2006), 125-135 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
[DoHu1]DH A. Douady, J. H. Hubbard, - 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**
[He]He M. Herman, - 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
[Ro]Ro P. Roesch, - 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
[Sh]Sh M. Shishikura, - 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
[Za]Za S. Zakeri ,

*Etude dynamique des polynômes complexes*, Publications mathématiques d’Orsay 1984.

*Conjugaison quasi-symétrique des difféomorphismes du cercle à desrotations et applications aux disques singuliers de Siegel, I.*, manuscript, http://www. math.kyoto-u.ac.jp/mitsu/Herman/qsconj2

*On local connectivity for the Julia set of rational maps*, Ann. Math. (To appear).

*The connectivity of the Julia set of rational maps and Fixed points*, Preprint IHES, Bures-sur-Yvette, 1992.

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

## Additional Information

**P. Roesch**- Affiliation: UMR Paul Painleve, University of Lille 1, Cité scientifique - Bâtiment M2, 69655 Villeneuve d’Ascq Cedex, France
- Email: roesch@math.univ-lille1.fr
- Received by editor(s): May 11, 2005
- Received by editor(s) in revised form: April 7, 2006
- Published electronically: July 6, 2006
- Additional Notes: Research partially supported by the Morningside Center of Mathematics in Beijing
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn.
**10**(2006), 125-135 - MSC (2000): Primary 37F50; Secondary 37F10
- DOI: https://doi.org/10.1090/S1088-4173-06-00139-1
- MathSciNet review: 2237276