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Title: An equivalent system for studying periodic points of the beta-transformation for a Pisot or a Salem number
Author: Maia, Bruno Henrique Prazeres de Melo e
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2007
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Abstract:
We propose an equivalent system (G,L) for studying the set of eventually periodic points, Per(T{3) , for the beta-transformation 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 semi-conjugate and furthermore the semi-conjugacy is one-to-one. 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 beta-expansions 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: University of Warwick, 2007 Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.490680  DOI: Not available
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