Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2012-11171-2
Published electronically: February 3, 2012
MathSciNet review: 2929011
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A. Douady and J. H. Hubbard.
    Exploring the Mandelbrot set. The Orsay notes.
    www.math.cornell.edu/~hubbard/OrsayEnglish.pdf.
  • 2. A. Douady and J. H. Hubbard.
    A proof of Thurston's topological characterization of rational functions.
    Acta Math., 171(2):263-297, 1993. MR 1251582 (94j:58143)
  • 3. A. Epstein.
    Integrality and rigidity for postcritically finite polynomials, Bull. London Math. Soc., 2011, doi:10.1112/blms/bdr059.
  • 4. G. Levin.
    Multipliers of periodic orbits in spaces of rational maps.
    Ergodic Theory and Dynamical Systems, 31(01): 197-243, 2011. MR 2755929
  • 5. C. T. McMullen and D. P. Sullivan.
    Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system.
    Adv. Math., 135(2):351-395, 1998. MR 1620850 (99e:58145)
  • 6. J. Milnor.
    Cubic polynomial maps with periodic critical orbit. I.
    Complex Dynamics.
    A K Peters, Wellesley, MA, 2009: 333-411. MR 2508263
  • 7. J. H. Silverman.
    The Arithmetic of Dynamical Systems, volume 241 of Graduate Texts in Mathematics.
    Springer, New York, 2007. MR 2316407 (2008c:11002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37F10, 37P05, 37P45

Retrieve articles in all journals with MSC (2010): 37F10, 37P05, 37P45


Additional Information

Joseph H. Silverman
Affiliation: Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
Email: jhs@math.brown.edu

DOI: https://doi.org/10.1090/S0002-9939-2012-11171-2
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.

American Mathematical Society