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Nielsen-Thurston reducibility and renormalization

Authors: Olivier Courcelle, Jean-Marc Gambaudo and Charles Tresser
Journal: Proc. Amer. Math. Soc. 125 (1997), 3051-3058
MSC (1991): Primary 58F99
MathSciNet review: 1425117
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Abstract: Consider an orientation preserving homeomorphism $f$ of the 2-disk with an infinite set of nested periodic orbits $\{\mathcal {O}_n\}_{n\ge 1}$, such that, for all $m>1$, the restriction of $f$ to the complement of the first $m$ orbits, from $\mathcal {O}_1$ to $\mathcal {O}_m$, is $m-1$ times reducible in the sense of Nielsen and Thurston. We define combinatorial renormalization operators for such maps, and study the fixed points of these operators. We also recall the corresponding theory for endomorphisms of the interval, and give elements of comparison of the theories in one and two dimensions.

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Additional Information

Olivier Courcelle
Affiliation: Section de Mathématiques, Université de Genève, CP240, CH1211 Genève 24, Suisse

Jean-Marc Gambaudo
Affiliation: INLN, 1361 route des lucioles, Sophia-Antipolis, 06560 Valbonne, France

Charles Tresser
Affiliation: IBM, P.O. Box 218, Yorktown Heights, New York 10598
MR Author ID: 174225

Received by editor(s): October 24, 1995
Communicated by: Linda Keen
Article copyright: © Copyright 1997 American Mathematical Society