By Valérie Berthé, Michel Rigo

Across the world recognized researchers examine constructing traits in combinatorics with functions within the learn of phrases and in symbolic dynamics. They clarify the real options, offering a transparent exposition of a few fresh effects, and emphasise the rising connections among those varied fields. subject matters comprise combinatorics on phrases, trend avoidance, graph idea, tilings and thought of computation, multidimensional subshifts, discrete dynamical platforms, ergodic concept, numeration platforms, dynamical arithmetics, automata idea and synchronised phrases, analytic combinatorics, persisted fractions and probabilistic versions. each one subject is gifted in a manner that hyperlinks it to the most topics, yet then also they are prolonged to repetitions in phrases, similarity kin, mobile automata, friezes and Dynkin diagrams. The e-book will entice graduate scholars, examine mathematicians and laptop scientists operating in combinatorics, idea of computation, quantity conception, symbolic dynamics, tilings and stringology. it is going to additionally curiosity biologists utilizing textual content algorithms.

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**Combinatorics, Words and Symbolic Dynamics**

Across the world regarded researchers examine constructing tendencies in combinatorics with purposes within the examine of phrases and in symbolic dynamics. They clarify the real innovations, offering a transparent exposition of a few contemporary effects, and emphasise the rising connections among those various fields.

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I=1 Loosely speaking, the greedy algorithm chooses in each step the largest possible digit. Let us show that (bi ) is an expansion of x for each x belonging to Jβ . If x = 1/(β − 1), then bi = 1 for each i ≥ 1, whence (bi ) is indeed an expansion of x. If 0 ≤ x < 1/(β − 1), then bn = 0 for some index n and for each such n we have ∞ bi 1 1 < n ≤ ∑ i. i β β β i=1 i=n+1 n 0 ≤ x− ∑ Hence there cannot be a last n such that bn = 0. Letting n → ∞ along the indices n for which bn = 0, we see that (bi ) is indeed an expansion of x.

Hence none of the numbers cos(2πβ m ) equals zero. 8) m=M > 0, −m conwhere the last inequality follows from the fact that the series ∑∞ m=M 2πβ verges. Similarly, we may choose M such that 2π kα M < 1. 9) > 0. 9). , absolutely continuous or singular) of the inﬁnite Bernoulli convolutions νβ in terms of the numbers Nn (x, β ). This enabled him to determine √ the growth rate of the numbers Nn (x, β ) as n → ∞ for almost every β ∈ (1, 2) and for almost every x ∈ Jβ . In order to present his main results, we note that (x1 , .

2 may be rewritten as Nβ = x ∈ Jβ |∃ω ∈ Ω such that Kβn (ω , x) ∈ Ω × S for only ﬁnitely many n ≥ 0 . Hence if x ∈ Jβ \ Nβ , then (di (ω , x)) = (di (ω , x)) whenever ω = ω . 7 Almost every x ∈ Jβ has a continuum of universal expansions. Proof Fix a block i1 · · · in ∈ {0, 1}n . , C = {ω ∈ Ω | ω1 = j1 ,... , ωn = jn } for some j1 ,... , jn ∈ {0,1}. 9. 31 Expansions in non-integer bases (ω , x) ∈ Ω × Jβ , the block i1 · · · in occurs in (di (ω , x)) with positive limiting relative frequency m ⊗ μβ (ω , x) ∈ Ω × Jβ | d1 (ω , x) = i1 , .