An algebraic formulation of Thurston’s characterization of rational functions

Kevin M. Pilgrim[1]

  • [1] Dept. of Mathematics, Indiana University, Bloomington, IN, 47405 USA

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: S5, page 1033-1068
  • ISSN: 0240-2963

Abstract

top
Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston’s characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichmüller space used in the proof of the characterization theorem.

How to cite

top

Pilgrim, Kevin M.. "An algebraic formulation of Thurston’s characterization of rational functions." Annales de la faculté des sciences de Toulouse Mathématiques 21.S5 (2012): 1033-1068. <http://eudml.org/doc/251019>.

@article{Pilgrim2012,
abstract = {Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston’s characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichmüller space used in the proof of the characterization theorem.},
affiliation = {Dept. of Mathematics, Indiana University, Bloomington, IN, 47405 USA},
author = {Pilgrim, Kevin M.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Thurston map; multicurve; obstruction; rational map; skinning map; mapping class group; virtual endomorphism},
language = {eng},
month = {12},
number = {S5},
pages = {1033-1068},
publisher = {Université Paul Sabatier, Toulouse},
title = {An algebraic formulation of Thurston’s characterization of rational functions},
url = {http://eudml.org/doc/251019},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Pilgrim, Kevin M.
TI - An algebraic formulation of Thurston’s characterization of rational functions
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/12//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - S5
SP - 1033
EP - 1068
AB - Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston’s characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichmüller space used in the proof of the characterization theorem.
LA - eng
KW - Thurston map; multicurve; obstruction; rational map; skinning map; mapping class group; virtual endomorphism
UR - http://eudml.org/doc/251019
ER -

References

top
  1. Buff (X.), Epstein (A.), Koch (S.), and Pilgrim (K. M.).— On Thurston’s pullback map. Complex dynamics, p. 561-583, A K Peters, Wellesley, MA (2009). Zbl1180.37065MR2508269
  2. Bartholdi (L.) and Nekrashevych (V.).— Thurston equivalence of topological polynomials. Acta Math.197, p. 1-51 (2006). Zbl1176.37020MR2285317
  3. Bonk (M.) and Meyer (D.).— Expanding Thurston maps. arXiv:1009.3647v1 (2010). 
  4. Brock (J.) and Margalit (D.).— Weil-Petersson isometries via the pants complex. Proc. Amer. Math. Soc. 1353, p. 795-803 (2007). Zbl1110.32004MR2262875
  5. Douady (A.) and Hubbard (J.).— A Proof of Thurston’s Topological Characterization of Rational Functions. Acta. Math.171, p. 263-297 (1993). Zbl0806.30027MR1251582
  6. Dym (H.).— Linear algebra in action, volume 78 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2007). Zbl1113.15001MR2289254
  7. Farb (B.) and Margalit (D.).— A primer on mapping class groups. Princeton University Press, Princeton, NJ. xiv+472 pp. ISBN: 978-0-691-14794-9 (2012). Zbl1245.57002MR2850125
  8. Kelsey (G.).— Mapping schemes realizable by obstructed topological polynomials, Conformal Geometry and Dynamics (electronic) 16, p. 44-80 (2012). Zbl1278.37043MR2893472
  9. Kent (R.).— Skinning maps. Duke Math. J.1512, p. 279-336 (2010). Zbl1193.30062MR2598379
  10. Koch (S.).— A new link between Teichmüller theory and complex dynamics. PhD thesis, Cornell University (2008). http://www.math.harvard.edu/~kochs/papers.html. 
  11. Lodge (R.).— The boundary values of Thurston’s pullback map. PhD thesis, Indiana University (2012). http://mypage.iu.edu/~rlodge/LodgeThesisFinal.pdf. 
  12. Milnor (J.) and Thurston (W.).— On iterated maps of the interval. Springer Lecture Notes in Mathematics 1342 (1988). Zbl0664.58015MR970571
  13. Nekrashevych (V.).— Self-similar groups, volume 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2005). Zbl1087.20032MR2162164
  14. Nekrashevych (V.).— Combinatorics of polynomial iterations. Complex dynamics, 169-214, A K Peters, Wellesley, MA (2009). Zbl1201.37078MR2508257
  15. Nielsen (J.).— Abbildungsklassen endlicher Ordnung. Acta Math.75, p. 23-115 (1943). Zbl0027.26601MR13306
  16. Pilgrim (K. M.).— Combinations of complex dynamical systems. Springer Lecture Notes in Mathematics 1827 (2003). Zbl1045.37028MR2020454
  17. Selinger (N.).— Thurston’s pullback map on the augmented Teichmüller space and applications, Inventiones Mathematicae 189, No. 1, p. 111-142 (2012). Zbl1298.37033MR2929084
  18. Thurston (W. P.).— On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.)19, p. 417-431 (1988). Zbl0674.57008MR956596
  19. Wolpert (S.).— The Weil-Petersson metric geometry. Handbook of Teichmüller theory, vol. II. IRMA Lect. Math. Theor. Phys., Eur. Math. Soc., Zürich (2009). Zbl1169.30020MR2497791

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.