Bijective automata

Abstract

In this chapter, we study a special case of automata, called bijective automata, whose instructions are permutations of the letters of A, including the case of additionally commutative automata in the previous sense. In order to describe the measures generating the maximal spectral type, we investigate the structure of the associated system with a new viewpoint. In the commutative bijective case, we give a complete description of those measures, and we touch on the spectral multiplicity problem which will be discussed later. In the non-commutative case, we only succeed in exhibiting strongly mixing generating measures. Note that the sequences arising from bijective substitutions are related to generalized Morse sequences studied in [156, 175, 176]. © 2010 Springer-Verlag Berlin Heidelberg.