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Substitutions in Dynamics, Arithmetics and Combinatorics by N. Pytheas Fogg

By N. Pytheas Fogg

A sure class of limitless strings of letters on a finite alphabet is gifted the following, selected one of the 'simplest' attainable one may possibly construct, either simply because they're very deterministic and since they're equipped by way of basic ideas (a letter is changed via a observe, a chain is produced by way of iteration). those substitutive sequences have a shockingly wealthy structure.
The authors describe the strategies of volume of typical interactions, with combinatorics on phrases, ergodic concept, linear algebra, spectral idea, geometry of tilings, theoretical desktop technological know-how, diophantine approximation, trancendence, graph idea. This quantity fulfils the necessity for a reference at the simple definitions and theorems, in addition to for a cutting-edge survey of the tougher and unsolved problems.

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5. 2. Substitutions, arithmetic and finite automata: an introduction The aim of this chapter is to introduce substitutions by showing some typical and important examples of situations in number theory where they appear. Special stress will be given to the statistical properties of these sequences. We first recall some properties of the Morse sequence, then we introduce the Rudin-Shapiro sequence and focus on its spectral properties. We also evoke the Baum-Sweet sequence, the Cantor sequence and the Fibonacci sequence.

We can deduce from the equality σ r+1 (a) = σ(σ r (a)) = σ r (σ(a)), the following combinatorial properties of the Morse sequence u: • ∀n ∈ N, u2n = un and u2n+1 = 1 − un ; • at position k2n of the sequence occurs σ n (a), if uk = 0, and σ n (b) if uk = 1. Let Ur = σ r (a) and Ur = σ r (b). These sequences of words over the alphabet {a, b} are uniquely defined by the following relation: 1 This chapter has been written by C. Mauduit N. Pytheas Fogg: LNM 1794, V. Berth´e et al. ), pp. 35–52, 2002.

These are usually called rotations of the group G. This class contains in particular all the toral rotations, the additions over a finite group, translations over p-adic integer groups or translations over p-adic solenoids (see Sec. 2). 4. Let G be a compact metric group. Let T : G → G, x → ax be a rotation of G. Prove that (G, T ) is minimal if and only if {an , n ∈ N} is dense in X. 5. The rotation by the vector α = (α1 , . . , αd ) ∈ Rd on the d-dimensional torus Td = Rd /Zd is minimal if and only if α1 , .

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