AUTOMATIC MAPS IN EXOTIC NUMERATION SYSTEMS

We generalize the classical notion of a b-automatic sequence for a seq uence indexed by the natural numbers. We replace the integers by a sem iring and use a numeration system consisting of the powers of a base b and an appropriate set of digits. For example, we define (-3)-automat ic sequences (indexed by the ordinary integers or by the rational inte gers) and (-1 + i)-automatic sequences (indexed by the Gaussian intege rs). We show how these new notions are related to the old ones, and we study both the number-theoretic and automata-theoretic properties tha t permit the replacement of one numeration system by another.