Title:

An equivalent system for studying periodic points of the betatransformation for a Pisot or a Salem number

We propose an equivalent system (G,L) for studying the set of eventually periodic
points, Per(T{3) , for the betatransformation of the unit interval, when {3 is a Pisot or
a Salem number. This system' is defined by a map G, which is closely related to the
companion matrix C of the minimal polynomial of {3 (of degree d 2: 2), and by a set of
points L C Qd.
The systems (G, L) and (T{3, [0, 1) nQ({3)) are semiconjugate and furthermore
the semiconjugacy is onetoone. Given that Per(T{3) ~ [0,1) n Q({3), we say that
(G,L) is an equivalent system as far as the study of periodic points is concerned.
We define symbolic dynamics for (G, L), which is related to the betaexpansions
of numbers in the unit interval. We show that Gcan be factored to the toral automorphism
defined by C and we also study the geometry of (G, L).
The main motivation for this work is Schmidt's paper [Sch80], and in particular
the theorem that Per(T{3) = [0, 1)nQ({3) when {3 is a Pisot number, and the conjecture
that the same should be true when {3 is a Salem number. We compare the different
dynamical behaviours of (G, L) when {3 is Pisot and when {3 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.
