## Prefix-suffix automaton associated with a primitive substitution. (Automate des préfixes-suffixes associé à une substitution primitive.)(French)Zbl 1071.37011

This paper deals with substitutive dynamical systems. It is proved that a dynamical system $$\Omega$$ arising from a primitive substitution is measurably conjugate to an adic transformation (in the sense of A. M. Vershik) on a subshift of finite type. This subshift is defined as the set of paths on a graph called the prefix-suffix automaton and which has been introduced in a weaker form by G. Rauzy for instance in [Sequences defined by iterated morphisms. Sequences, combinatorics, compression, and transmission, Pap. Adv. Int. Workshop, Naples/Italy 1988, 275–286 (1990; Zbl 0955.28501)]. The authors prove that the conjugation map is one-to-one except on the orbit of periodic points of $$\Omega$$, on which it is finite-to-one. They deduce a sequence of partitions of $$\Omega$$ which is generating in measure. This work is completed by the other article of the authors [Trans. Am. Math. Soc. 353, No. 12, 5121–5144 (2001; Zbl 1142.37302)] where this approach allows them to obtain geometric representations for some of these systems as irrational translations on tori.

### MSC:

 37B15 Dynamical aspects of cellular automata 11B85 Automata sequences 28A80 Fractals 28D05 Measure-preserving transformations 37B10 Symbolic dynamics

### Citations:

Zbl 0955.28501; Zbl 1142.37302
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### References:

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