Confluent alternate numeration systems

Inspired by methods of Masáková, Pelantová and Studeničová, we study normalisation in alternate base numeration systems through the lens of rewriting rules. We show that normalisation is only possible in this fashion with a confluent system of rewriting rules and give a characterisation of this property based on the expansions of 1. The same characterisation is then linked to the spectrum of the numeration system.