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

• Author(s): Maia, Bruno
• Date: 2017
• Persistent ID: http://hdl.handle.net/11144/3119
• Origin: Camões - Repositório Institucional da Universidade Autónoma de Lisboa
• Subject(s): Beta-transformation; companion matrix; Pisot; Salem; periodic orbit
• Document Type: Journal article
• Language: English

Description

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.