By Holton D.A. (ed.)

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In particular, the number of words of length n ≥ 0 over {0, 1} not containing the factor “11” is Fn + Fn−1 = Fn+1 . Indeed, one has to count paths of length n from the initial state to one of the two terminal states. So we sum up the first two entries on the first row. The same kind of idea can be applied to morphisms. Let σ : A∗ → A∗ be an endomorphism. The matrix Mσ ∈ NA×A associated with σ is called the incidence matrix of σ and is defined by ∀a, b ∈ A, (Mσ )a,b = |σ(b)|a . Let us recall that P stands for the abelianisation map.

8 Let σ be a primitive substitution. 19. We denote it by (Xσ , S). Preliminaries 47 Let us quote an interesting property of substitutive dynamical systems associated with a primitive substitution. For more details, see (Queff´elec 1987) and (Pytheas Fogg 2002). 9 Let σ be a primitive substitution. The system (Xσ , S) is uniquely ergodic. 5. 1 Show that over a binary alphabet, any word of length ≥ 4 contains a square as factor. 2 Show that over a ternary alphabet A, it is possible to build an infinite word avoiding squares.

Fig. 5. Two redundancy transducers. † It has been given a renewed interest in computer arithmetic for the redundancy † It was known (at least) as early as Cauchy who advocated such system for p = 10 and h = 5 with the argument that it makes the learning of addition and multiplication easier: the size of the tables is rougly divided by 4 (see (Cauchy 1840)). 60 Ch. Frougny, J. Sakarovitch in the representations allows to improve the way operations are performed, as we shall see now. The following is to be found in (Avizienis 1961) for bases larger than 2, in (Chow and Robertson 1978) for the binary case — although the original statements and proofs are not formulated in terms of automata.