Maximal Spectral Type of General Automata

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.