Linear numeration systems and \(\theta\)-representations (Q1190473)

The author considers bases of numeration given by a linear recurrence with integral coefficients. She is interested in ``the normalization function'' which maps any representation of an integer to its ``normal'' representation (i.e. the greedy algorithm representation). She gives a property of the base which ensures that the normalization function can be computed by a finite automaton. In a second part she studies the same problem for the representation of real numbers in base \(\theta\) (\(\theta\) a real number \(>1\)), in connection with symbolic dynamics: in particular she proves that any normalization function can be computed by a finite automaton if \(\theta\) is a Pisot number. [Note that the author and \textit{D. Berend} obtained very recently that the reciprocal of this theorem holds true].