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Maximal Spectral Type of General Automata

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1294)

Abstract

In this chapter, we prove the main theorem relative to the spectral study of primitive and aperiodic substitutions of length q or q-automata. Since only the continuous part of the spectrum has to be described, we may restrict our attention to pure substitutions without loss of generality (according to item 6.3.1.2); we shall get the following : the maximal spectral type is generated by k ≤ s probability measures which are strongly mixing with respect to the q-shift S q (in case the height is one). To get this result, we associate to the substitution ζ, a new substitution defined on A×A, the alphabet consisting in pairs of letters of A, and whose correlation matrix always enjoys wonderful properties.

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Correspondence to Martine Queffélec .

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© 2010 Springer-Verlag Berlin Heidelberg

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Queffélec, M. (2010). Maximal Spectral Type of General Automata. In: Substitution Dynamical Systems - Spectral Analysis. Lecture Notes in Mathematics(), vol 1294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11212-6_10

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