Jump to ContentJump to Main Navigation
Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
Users without a subscription are not able to see the full content.

Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 21 September 2021

The quadratic dynatomic curves are smooth and irreducible

The quadratic dynatomic curves are smooth and irreducible

Chapter:
(p.49) The quadratic dynatomic curves are smooth and irreducible
Source:
Frontiers in Complex Dynamics
Author(s):

Xavier Buff

Tan Lei

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0005

This chapter re-proves both the smoothness and the irreducibility of the quadratic dynatomic curves {(c,z) ∈ C² ∣ z is n-periodic for z² + c}. The smoothness is due to previous work laid out by Douady–Hubbard. Moreover, the proof here is based on elementary calculations on the pushforwards of specific quadratic differentials. This approach is a computational illustration of the power of the far more general transversality theory of A. Epstein, and is further inspired by the proof of Lau-Schleicher. The irreducibility is due to Bousch and Lau-Schleicher, but with a different method. Finally, the chapter uses elementary combinatorial properties of the kneading sequences instead of internal addresses.

Keywords:   quadratic dynatomic curves, smoothness, pushforwards, quadratic differentials, transversality theory, irreducibility, kneading sequences

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.