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Fixed-point-free elements of iterated monodromy groups. (English) Zbl 1385.37063
Summary: 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.

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
20E08 Groups acting on trees
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
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[1] Bartholdi, Laurent; Nekrashevych, Volodymyr, Thurston equivalence of topological polynomials, Acta Math., 197, 1, 1-51, (2006) · Zbl 1176.37020
[2] Bartholdi, Laurent; Nekrashevych, Volodymyr, Iterated monodromy groups of quadratic polynomials. I, Groups Geom. Dyn., 2, 3, 309-336, (2008) · Zbl 1153.37379
[3] Bartholdi, Laurent; Vir\'ag, B\'alint, Amenability via random walks, Duke Math. J., 130, 1, 39-56, (2005) · Zbl 1104.43002
[4] Bondarenko, E.; Nekrashevych, V., Post-critically finite self-similar groups, Algebra Discrete Math., 4, 21-32, (2003) · Zbl 1068.20028
[5] [forster] Otto Forster. \newblockLectures on Riemann surfaces, volume 81 of Graduate Texts in Mathematics. \newblock Springer-Verlag, New York, 1991. \newblock Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation.
[6] Grigorchuk, Rostislav; Savchuk, Dmytro; Suni\'c, Zoran, The spectral problem, substitutions and iterated monodromy. Probability and mathematical physics, CRM Proc. Lecture Notes 42, 225-248, (2007), Amer. Math. Soc.: Providence, RI:Amer. Math. Soc. · Zbl 1138.20025
[7] Grimmett, Geoffrey R.; Stirzaker, David R., Probability and random processes, xii+596 pp., (2001), Oxford University Press: New York:Oxford University Press · Zbl 1015.60002
[8] Jones, Rafe, Iterated Galois towers, their associated martingales, and the \(p\)-adic Mandelbrot set, Compos. Math., 143, 5, 1108-1126, (2007) · Zbl 1166.11040
[9] Jones, Rafe, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2), 78, 2, 523-544, (2008) · Zbl 1193.37144
[10] Kahn, Jeremy; Lyubich, Mikhail; Rempe, Lasse, A note on hyperbolic leaves and wild laminations of rational functions, J. Difference Equ. Appl., 16, 5-6, 655-665, (2010) · Zbl 1198.30031
[11] Lyubich, Mikhail; Minsky, Yair, Laminations in holomorphic dynamics, J. Differential Geom., 47, 1, 17-94, (1997) · Zbl 0910.58032
[12] Makarov, N.; Smirnov, S., Phase transition in subhyperbolic Julia sets, Ergodic Theory Dynam. Systems, 16, 1, 125-157, (1996) · Zbl 0852.58067
[13] Makarov, N.; Smirnov, S., On “thermodynamics” of rational maps. I. Negative spectrum, Comm. Math. Phys., 211, 3, 705-743, (2000) · Zbl 0983.37033
[14] Milnor, John, Dynamics in one complex variable, Annals of Mathematics Studies 160, viii+304 pp., (2006), Princeton University Press: Princeton, NJ:Princeton University Press · Zbl 1085.30002
[15] Nekrashevych, Volodymyr, Self-similar groups, Mathematical Surveys and Monographs 117, xii+231 pp., (2005), American Mathematical Society: Providence, RI:American Mathematical Society · Zbl 1087.20032
[16] Nekrashevych, Volodymyr, Iterated monodromy groups. Groups St Andrews 2009 in Bath. Volume 1, London Math. Soc. Lecture Note Ser. 387, 41-93, (2011), Cambridge Univ. Press: Cambridge:Cambridge Univ. Press · Zbl 1235.37016
[17] Rosen, Michael, Number theory in function fields, Graduate Texts in Mathematics 210, xii+358 pp., (2002), Springer-Verlag: New York:Springer-Verlag · Zbl 1043.11079
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