An algebraic approach to certain cases of Thurston rigidity
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- by Joseph H. Silverman PDF
- Proc. Amer. Math. Soc. 140 (2012), 3421-3434 Request permission
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 $1$.References
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Additional Information
- Joseph H. Silverman
- Affiliation: Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
- MR Author ID: 162205
- ORCID: 0000-0003-3887-3248
- Email: jhs@math.brown.edu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3421-3434
- MSC (2010): Primary 37F10; Secondary 37P05, 37P45
- DOI: https://doi.org/10.1090/S0002-9939-2012-11171-2
- MathSciNet review: 2929011