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An algebraic approach to certain cases of Thurston rigidity

Author: Joseph H. Silverman
Journal: Proc. Amer. Math. Soc. 140 (2012), 3421-3434
MSC (2010): Primary 37F10; Secondary 37P05, 37P45
Published electronically: February 3, 2012
MathSciNet review: 2929011
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Abstract: In the parameter space of monic centered polynomials of degree $ 3$ with marked critical points $ c_1$ and $ c_2$, let $ C_{1,n}$ be the locus of maps for which $ c_1$ has period $ n$ and let $ C_{2,m}$ be the locus of maps for which $ c_2$ has period $ m$. A consequence of Thurston's rigidity theorem is that the curves $ C_{1,n}$ and $ C_{2,m}$ intersect transversally. We give a purely algebraic proof that the intersection points are $ 3$-adically integral and use this to prove transversality. We also prove an analogous result when $ c_1$ or $ c_2$ or both are taken to be preperiodic with tail length exactly $ 1$.

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Additional Information

Joseph H. Silverman
Affiliation: Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912

Received by editor(s): October 21, 2010
Received by editor(s) in revised form: April 5, 2011
Published electronically: February 3, 2012
Additional Notes: The author’s research is supported by NSF DMS-0650017 and DMS-0854755.
Communicated by: Bryna Kra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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