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

By N. Pytheas Fogg

A yes class of endless strings of letters on a finite alphabet is gifted right here, selected one of the 'simplest' attainable one could construct, either simply because they're very deterministic and since they're outfitted by means of basic ideas (a letter is changed by means of a observe, a series is produced through iteration). those substitutive sequences have a shockingly wealthy structure.
The authors describe the suggestions of volume of traditional interactions, with combinatorics on phrases, ergodic idea, linear algebra, spectral thought, geometry of tilings, theoretical laptop technology, 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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Extra info for Substitutions in Dynamics, Arithmetics and Combinatorics

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1. Prove that the addition of any coordinate ak on T is also minimal. 2. Prove that the addition of a on S2 is onto. 3. Prove that the addition of a on S2 is minimal. 6 Factors of substitutive dynamical systems 31 Let us finally study some spectral properties of the minimal addition of a on the 2-adic solenoid. 1. Prove that the spectrum of the addition of a on S2 contains all the numbers e2iπak , k ∈≥ 1, and that it is exactly equal to the group generated by mα ∪k≥1 exp(2iπZ[α/2k ]) = {e2iπ 2k , k ≥ 1, m ∈ Z}.

2 that Zp is a measure-theoretic factor of (X, T, µ). The 2-adic solenoid. Solenoids usually appear in dynamical systems as examples of attractors (see for instance the Smale attractor in [234], or [140]). See also [247] for a purely symbolic approach in connection with the notion of dimension group. We will study some properties of the 2-adic solenoid solenoid through an exercise. Everything extends in a natural way to the p-adic solenoid. The 2-adic solenoid S2 can be realized as the following inverse limit space: S2 = {x = (xk )k≥1 | ∀k ≥ 1, xk ∈ T, xk = 2xk+1 mod 1} .

Since the space C(X) is metrizable, it is equivalent to say that there exists an increasing sequence of integers (Nk )k∈N such that ∀f ∈ C(X), 1 Nk f (T n x) → n

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