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Title: The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits
Authors: Maia, Bruno
Keywords: Beta-transformation
companion matrix
periodic orbit
Issue Date: 2017
Publisher: Taylor & Francis
Abstract: 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.
Peer Reviewed: yes
metadata.dc.identifier.doi: 10.1080/14689367.2017.1288701
ISSN: 1468-9367
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Appears in Collections:DCEE - Artigos/Papers

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