The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits

Abstract

ABSTRACTThe β-transformation of the unit interval is defined by Tβ(x) ≔ βx (mod 1). Its eventually periodic points are a subset of [0, 1] intersected with the field extension . If β > 1 is an algebraic integer of degree d > 1, then is a -vector space isomorphic to ; therefore, the intersection of [0, 1] with is isomorphic to a domain in . The transformation from this domain which is conjugate to the β-transformation is called the companion map, given its connection to the companion matrix of β's minimal polynomial. The companion map and the proposed notation provide a natural setting to reformulate a classic result concerning the set of periodic points of the β-transformation for numbers. It also allows to visualize orbits in a d-dimensional space. Finally, we refer connections with arithmetic codings and symbolic representations of hyperbolic toral automorphisms.