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Fixed-point-free elements of iterated monodromy groups

Author: Rafe Jones
Journal: Trans. Amer. Math. Soc. 367 (2015), 2023-2049
MSC (2010): Primary 37F10, 20E08; Secondary 37P05, 37F20
Published electronically: October 1, 2014
MathSciNet review: 3286507
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Abstract: The iterated monodromy group of a post-critically finite complex polynomial of degree $ d \geq 2$ acts naturally on the complete $ d$-ary rooted tree $ T$ of preimages of a generic point. This group, as well as its profinite completion, acts on the boundary of $ T$, which is given by extending the branches to their ``ends'' at infinity. We show that, in most cases, elements that have fixed points on the boundary are rare, in that they belong to a set of Haar measure 0. The exceptions are those polynomials linearly conjugate to multiples of Chebyshev polynomials and a case that remains unresolved, where the polynomial has a non-critical fixed point with many critical preimages. The proof involves a study of the finite automaton giving generators of the iterated monodromy group, and an application of a martingale convergence theorem. Our result is motivated in part by applications to arithmetic dynamics, where iterated monodromy groups furnish the ``geometric part'' of certain Galois extensions encoding information about densities of dynamically interesting sets of prime ideals.

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Rafe Jones
Affiliation: Department of Mathematics, Carleton College, One North College Street, Northfield, Minnesota 55057

Received by editor(s): June 9, 2012
Received by editor(s) in revised form: May 1, 2013
Published electronically: October 1, 2014
Additional Notes: The author’s research was partially supported by NSF grant DMS-0852826
Article copyright: © Copyright 2014 American Mathematical Society

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