Introduction to Iterated Monodromy Groups

Sébastien Godillon[1]

  • [1] Laboratoire de Mathématiques (UMR 8088), Université de Cergy-Pontoise, F-95000 Cergy-Pontoise, France

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

  • Volume: 21, Issue: S5, page 1069-1118
  • ISSN: 0240-2963

Abstract

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The theory of iterated monodromy groups was developed by Nekrashevych [9]. It is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. Iterated monodromy groups encode in a computationally efficient way combinatorial information about any dynamical system induced by a post-critically finite branched covering. Their power was illustrated by a solution of the Hubbard Twisted Rabbit Problem given by Bartholdi and Nekrashevych [2].These notes attempt to introduce this theory for those who are familiar with holomorphic dynamics but not with group theory. The aims are to give all explanations needed to understand the main definition (Definition 3.6) and to provide skills in computing any iterated monodromy group efficiently (see examples in Section 3.3). Moreover some explicit links between iterated monodromy groups and holomorphic dynamics are detailed. In particular, Section 4.1 provides some facts about combinatorial equivalence classes, and Section 4.2 deals with matings of polynomials.

How to cite

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Godillon, Sébastien. "Introduction to Iterated Monodromy Groups." Annales de la faculté des sciences de Toulouse Mathématiques 21.S5 (2012): 1069-1118. <http://eudml.org/doc/250994>.

@article{Godillon2012,
abstract = {The theory of iterated monodromy groups was developed by Nekrashevych [9]. It is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. Iterated monodromy groups encode in a computationally efficient way combinatorial information about any dynamical system induced by a post-critically finite branched covering. Their power was illustrated by a solution of the Hubbard Twisted Rabbit Problem given by Bartholdi and Nekrashevych [2].These notes attempt to introduce this theory for those who are familiar with holomorphic dynamics but not with group theory. The aims are to give all explanations needed to understand the main definition (Definition 3.6) and to provide skills in computing any iterated monodromy group efficiently (see examples in Section 3.3). Moreover some explicit links between iterated monodromy groups and holomorphic dynamics are detailed. In particular, Section 4.1 provides some facts about combinatorial equivalence classes, and Section 4.2 deals with matings of polynomials.},
affiliation = {Laboratoire de Mathématiques (UMR 8088), Université de Cergy-Pontoise, F-95000 Cergy-Pontoise, France},
author = {Godillon, Sébastien},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {12},
number = {S5},
pages = {1069-1118},
publisher = {Université Paul Sabatier, Toulouse},
title = {Introduction to Iterated Monodromy Groups},
url = {http://eudml.org/doc/250994},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Godillon, Sébastien
TI - Introduction to Iterated Monodromy Groups
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/12//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - S5
SP - 1069
EP - 1118
AB - The theory of iterated monodromy groups was developed by Nekrashevych [9]. It is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. Iterated monodromy groups encode in a computationally efficient way combinatorial information about any dynamical system induced by a post-critically finite branched covering. Their power was illustrated by a solution of the Hubbard Twisted Rabbit Problem given by Bartholdi and Nekrashevych [2].These notes attempt to introduce this theory for those who are familiar with holomorphic dynamics but not with group theory. The aims are to give all explanations needed to understand the main definition (Definition 3.6) and to provide skills in computing any iterated monodromy group efficiently (see examples in Section 3.3). Moreover some explicit links between iterated monodromy groups and holomorphic dynamics are detailed. In particular, Section 4.1 provides some facts about combinatorial equivalence classes, and Section 4.2 deals with matings of polynomials.
LA - eng
UR - http://eudml.org/doc/250994
ER -

References

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  5. Grigorchuk (R. I.) and Żuk (A.).— On a torsion-free weakly branch group defined by a three state automaton. International Journal of Algebra and Computation, 12(1-2) p. 223-246 (2002). Zbl1070.20031MR1902367
  6. Levy (S. V. F.).— Critically finite rational maps. PhD thesis, Princeton University (1985). 
  7. Meyer (D.).— Unmating of rational maps, sufficient criteria and examples. arXiv:1110.6784v1 [math.DS] to appear in Proceeding for the Conference “Frontiers in Complex Dynamics (Celebrating John Milnor’s 80th birthday)" (2012). 
  8. Milnor (J.).— Remarks on quadratic rational maps. arXiv:math/9209221v1 [math.DS] (1992). MR1181083
  9. Nekrashevych (V.).— Self-similar groups, volume 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2005). Zbl1087.20032MR2162164
  10. Pilgrim (K. M.).— Dessins d’enfants and Hubbard trees. Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 33(5) p. 671-693 (2000). Zbl1066.14503MR1834499
  11. Pilgrim (K. M.).— An algebraic formulation of Thurston’s combinatorial equivalence. Proceedings of the American Mathematical Society, 131(11) p. 3527-3534 (2003). Zbl1113.37029MR1991765
  12. Pilgrim (K. M.).— Combinations of complex dynamical systems, volume 1827 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2003). Zbl1045.37028MR2020454
  13. Pilgrim (K. M.).— A Hurwitz-like classification of Thurston combinatorial classes. Osaka Journal of Mathematics, 41(1), p. 131-143 (2004). Zbl1083.57500MR2040069
  14. Lei (T.).— Matings of quadratic polynomials. Ergodic Theory and Dynamical Systems, 12(3) p. 589-620 (1992) Zbl0756.58024MR1182664

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