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

• Autor(a) principal: Maia, Bruno
• Data de Publicação: 2017
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/11144/3119
• topic: Beta-transformation; companion matrix; Pisot; Salem; periodic orbit

The -transformation of the unit interval is de ned by T (x) := x (mod 1). Its eventually periodic points are a subset of [0; 1] intersected with the eld extension Q( ). If > 1 is an algebraic integer of degree d > 1, then Q( ) is a Q-vector space isomorphic to Q d , therefore the intersection of [0; 1] with Q( ) is isomorphic to a domain in Q d . The transformation from this domain which is conjugate to the -transformation is called the companion map, given its connection to the companion matrix of '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 -transformation for Pisot 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.