## Periodic points and smooth rays

HTML articles powered by AMS MathViewer

- by Carsten Lunde Petersen and Saeed Zakeri PDF
- Conform. Geom. Dyn.
**25**(2021), 170-178 Request permission

## Abstract:

Let $P: \mathbb {C} \to \mathbb {C}$ be a polynomial map with disconnected filled Julia set $K_P$ and let $z_0$ be a repelling or parabolic periodic point of $P$. We show that if the connected component of $K_P$ containing $z_0$ is non-degenerate, then $z_0$ is the landing point of at least one*smooth*external ray. The statement is optimal in the sense that all but one cycle of rays landing at $z_0$ may be broken.

## References

- 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 - Lisa R. Goldberg and John Milnor,
*Fixed points of polynomial maps. II. Fixed point portraits*, Ann. Sci. École Norm. Sup. (4)**26**(1993), no. 1, 51–98. MR**1209913**, DOI 10.24033/asens.1667 - J. H. Hubbard,
*Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz*, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 467–511. MR**1215974** - G. Levin and F. Przytycki,
*External rays to periodic points*, Israel J. Math.**94**(1996), 29–57. MR**1394566**, DOI 10.1007/BF02762696 - John Milnor,
*Dynamics in one complex variable*, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR**2193309** - Carsten Lunde Petersen,
*On the Pommerenke-Levin-Yoccoz inequality*, Ergodic Theory Dynam. Systems**13**(1993), no. 4, 785–806. MR**1257034** - C. Petersen and S. Zakeri,
*On the correspondence of external rays under renormalization*, arXiv:1903.00800, 2019.

## Additional Information

**Carsten Lunde Petersen**- Affiliation: Department of Mathematics, Roskilde University, DK-4000 Roskilde, Denmark
- MR Author ID: 207550
- ORCID: 0000-0002-4890-3183
- Email: lunde@ruc.dk
**Saeed Zakeri**- Affiliation: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Blvd., Queens, New York 11367; and The Graduate Center of CUNY, 365 Fifth Ave., New York, NY 10016
- MR Author ID: 626230
- Email: saeed.zakeri@qc.cuny.edu
- Received by editor(s): February 26, 2021
- Received by editor(s) in revised form: July 25, 2021
- Published electronically: October 27, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**25**(2021), 170-178 - MSC (2020): Primary 37F10, 37F25, 37F20
- DOI: https://doi.org/10.1090/ecgd/364
- MathSciNet review: 4332105