Substitutive number systems

Abstract

In this paper, we associate a primitive substitution with a family of non-integer positional number systems with respect to the same base but with different sets of digits. In this way, we generalize the classical Dumont–Thomas numeration which corresponds to one specific case. Therefore, our concept also covers beta-expansions induced by Parry numbers. But we establish links to variants of beta-expansions such as symmetric beta-expansions, too. In other words, we unify several well-known notions of non-integer representations within one general framework. A focus in our research is set on finiteness and periodicity properties. It turns out that these characteristics mainly depend on the substitution. As a consequence we are able to relate known finiteness properties that are viewed independently yet.