Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Nearly Euclidean Thurston maps


Authors: J. W. Cannon, W. J. Floyd, W. R. Parry and K. M. Pilgrim
Journal: Conform. Geom. Dyn. 16 (2012), 209-255
MSC (2010): Primary 37F10, 37F20
DOI: https://doi.org/10.1090/S1088-4173-2012-00248-2
Published electronically: August 15, 2012
MathSciNet review: 2958932
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We take an in-depth look at Thurston's combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preserving branched maps $ f\colon S^2\to S^2$ whose local degree at every critical point is $ 2$ and which have exactly four postcritical points. These maps are simple enough to be tractable, but are complicated enough to have interesting dynamics.


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

  • 1. X. Buff, A. Epstein, S. Koch, and K. Pilgrim, On Thurston's pullback map, in Complex Dynamics: Family and Friends, Dierk Schleicher ed., Wellesley, 2009, 561-583, with erratum at arXiv:1105.1763. MR 2508269 (2010g:37071),
  • 2. J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153-196 (electronic). MR 1875951 (2002j:52021)
  • 3. A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), 263-297. MR 1251582 (94j:58143)
  • 4. B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Univ. Press, Princeton and Oxford, 2011. MR 2850125
  • 5. R. Lodge, Boundary values of the Thurston pullback map, Ph.D. Thesis, Indiana University, 2012.
  • 6. J. Milnor, Pasting together Julia sets: a worked out example of mating, Experimental Math. 13 (2004), 55-92. MR 2065568 (2005c:37087)
  • 7. J. Milnor, On Lattès maps, in Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich (2006), 9-43. MR 2348953 (2009h:37090)
  • 8. K. M. Pilgrim, An algebraic formulation of Thurston's characterization of rational functions, http://mypage.iu.edu/˜pilgrim/Research/Papers/Tw.pdf, to appear in Annales de la Faculté des Sciences de Toulouse.
  • 9. E. A. Saenz Maldonado, On nearly Euclidean Thurston maps, Ph.D. Thesis, Virginia Tech, 2012.
  • 10. N. Selinger, Thurston's pullback map on the augmented Teichmüller space and applications, Invent. Math. 189 (2012), no. 1, 111-142. MR 2929084
  • 11. W. P. Thurston, Lecture notes, CBMS Conference, University of Minnesota at Duluth, 1983.
  • 12. P. L. Walker, Elliptic Functions; a Constructive Approach, Wiley, Chichester, 1996. MR 1435743 (98g:33035)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 37F10, 37F20

Retrieve articles in all journals with MSC (2010): 37F10, 37F20


Additional Information

J. W. Cannon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: cannon@math.byu.edu

W. J. Floyd
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
Email: floyd@math.vt.edu

W. R. Parry
Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
Email: walter.parry@emich.edu

K. M. Pilgrim
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: pilgrim@indiana.edu

DOI: https://doi.org/10.1090/S1088-4173-2012-00248-2
Received by editor(s): April 16, 2012
Published electronically: August 15, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society