Positionality of Dumont-Thomas numeration systems

Dumont-Thomas numeration systems are a subclass of abstract numeration systems where the factorisation of the fixed point of a substitution is used to represent numbers. A positional numeration system is one where a weight can be assigned to each position so that the evaluation map is an inner product with the weights. For general abstract numeration systems, deciding positionality is an open problem. In this talk, we define an extension of Dumont-Thomas numeration systems to all integers. We then offer a criterion for deciding the positionality of such a system. If time permits, we show a link to Bertrand numeration systems, another familiar class of numeration systems.