An equivalent system for studying periodic points of the beta-transformation for a Pisot or a Salem number

Abstract

We propose an equivalent system ( e C, L) for studying the set of eventually periodic points, P er(T β ), for the beta-transformation of the unit interval, when β is a Pisot or a Salem number. This system is defined by a map e C, which is closely related to the companion matrix C of the minimal polynomial of β (of degree d ≥ 2), and by a set of points L ⊂ Q d . The systems ( e C, L) and T β , [0, 1) ∩ Q(β)

are semi-conjugate and furthermore the semi-conjugacy is one-to-one. Given that P er(T β ) ⊆ [0, 1) ∩ Q(β), we say that ( e C, L) is an equivalent system as far as the study of periodic points is concerned. We define symbolic dynamics for ( e C, L), which is related to the beta-expansions of numbers in the unit interval. We show that e C can be factored to the toral automorphism defined by C and we also study the geometry of ( e C, L). The main motivation for this work is Schmidt’s paper [Sch80], and in particular the theorem that P er(T β ) = [0, 1) ∩ Q(β) when β is a Pisot number, and the conjecture that the same should be true when β is a Salem number. We compare the different dynamical behaviours of ( e C, L) when β is Pisot and when β is Salem , and state some of the implications of Schmidt’s theorem and conjecture. Finally, we use computer simulations and plots for a particular Salem case of degree 4, with a view to gaining further insight about the general Salem case