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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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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

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The quadratic dynatomic curves are smooth and irreducible

The quadratic dynatomic curves are smooth and irreducible

(p.49) The quadratic dynatomic curves are smooth and irreducible
Frontiers in Complex Dynamics

Xavier Buff

Tan Lei

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Princeton University Press

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

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