On multiplicatively dependent linear numeration systems, and periodic points

Abstract

Abstract. Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

Number representation and finite automata

“…By showing that R θ does not satisfy an extended beta-polynomial, Theorem 2.3.57 implies that the set L(R θ ) is not recognisable by a finite automaton. A direct combinatorial proof can be found in (Frougny 2002).…”