The spectral problem, substitutions and
                    iterated monodromy

Grigorchuk, Rostislav; Savchuk, Dmytro; Sunic, Zoran

doi:10.1090/crmp/042/12

Cited by 17 publications

(24 citation statements)

References 12 publications

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“…The Schreier spectrum of IMG z 2 + i was considered in [GSŠ07], using the auxiliary 3-dimensional pencil of operators given by A (3) n (x, y, z) = a n + yb n + zc n − xI.…”

Section: Concrete Examples and Computation Resultsmentioning

confidence: 99%

From Self-Similar Groups to Self-Similar Sets and Spectra

Grigorchuk

Nekrashevych

Sunic

2015

Fractal Geometry and Stochastics V

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“…The Schreier spectrum of IMG z 2 + i was considered in [GSŠ07], using the auxiliary 3-dimensional pencil of operators given by A (3) n (x, y, z) = a n + yb n + zc n − xI.…”

Section: Concrete Examples and Computation Resultsmentioning

confidence: 99%

From Self-Similar Groups to Self-Similar Sets and Spectra

Grigorchuk

Nekrashevych

Sunic

2015

Fractal Geometry and Stochastics V

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“…This group is one more example of a group of intermediate growth (see [10]). The algebraic properties of IMG(z 2 + i) were studied in [17]. The Schreier graphs Γ w of this group have polynomial growth of degree log 2 log λ , where λ is the real root of x 3 −x−2 (see [6, Chapter VI]).…”

Section: Iterated Monodromy Group Of Z 2 + Imentioning

confidence: 99%

Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

D'Angeli

Bondarenko²,

Nagnibeda³

2017

J. Fractal Geom.

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Every self-similar group acts on the space X ω of infinite words over some alphabet X. We study the Schreier graphs Γ w for w ∈ X ω of the action of self-similar groups generated by bounded automata on the space X ω . Using sofic subshifts we determine the number of ends for every Schreier graph Γ w . Almost all Schreier graphs Γ w with respect to the uniform measure on X ω have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.

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“…Let P(f, t 0 ) be the set of primes dividing at least one element of the sequence f n (a 0 ) − t 0 , n ≥ 1. Suppose that f n (x) − t 0 is separable for all n ≥ 1, and let A ∞ (t 0 ) be as in (5). Then…”

Section: 2mentioning

confidence: 99%

Fixed-point-free elements of iterated monodromy groups

Jones¹

2014

Trans. Amer. Math. Soc.

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Abstract. The iterated monodromy group of a post-critically finite complex polynomial of degree d ≥ 2 acts naturally on the complete d-ary rooted tree T of preimages of a generic point. This group, as well as its pro-finite completion, act 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 pre-images. 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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The spectral problem, substitutions and iterated monodromy

Cited by 17 publications

References 12 publications

From Self-Similar Groups to Self-Similar Sets and Spectra

From Self-Similar Groups to Self-Similar Sets and Spectra

Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Fixed-point-free elements of iterated monodromy groups

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