Periodic representations in Salem bases

Abstract

We prove that all algebraic bases β allow an eventually periodic representation of the elements of ℚ(β) with a finite alphabet of digits $${\cal A}$$ . Moreover, the classification of bases allowing that those representations have the socalled weak greedy property is given. The decision problem whether a given pair (β, $${\cal A}$$ ) allows eventually periodic representations proves to be rather hard, for it is equivalent to a topological property of the attractor of an iterated function system.