Shift Radix Systems and Their Generalizations

Research output: ThesisDoctoral Thesis

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Abstract

The main subjects of this thesis are Shift Radix Systems (SRS) in different settings and the Schur-Cohn region and its generalizations. SRS were introduced by Akiyama, Borbély, Brunotte, Pethõ, and Thuswaldner in 2005 to generalize two important notions of number systems: Beta-expansions and Canonical Number Systems. Since their introduction SRS found great interest for their own sake and were subject of many publications. In this thesis new algorithms, characterization results, and topological results related to SRS as well as results on the Lebesgue measure of a generalized Schur-Cohn region are presented. Furthermore it includes related results on a special type of multiple integral due to Selberg and Aomoto. The thesis comes with a CD which contains an annotated version of the C++ program used to derive some of the presented results. The thesis contains six chapters which will be introduced in the following: Chapter 1, Selberg and Aomoto integrals: In this chapter certain types of integrals known as Selberg and Aomoto integrals are introduced and generalized. The formulas derived are needed in Chapter 2 to compute the volumes of parts of a certain subdivision of the Schur-Cohn region. Chapter 2, The Schur-Cohn region and its generalizations: The Schur-Cohn region is intimately related to a dynamical property of SRS. In this chapter previous results on the Schur-Cohn region are presented and a recent conjecture on the volumes of parts of a certain subdivision is treated and proved for a special case. Chapter 3, Shift Radix Systems and the finiteness property: In this chapter SRS and several important related tools and concepts are introduced. The relation to Beta-expansions and Canonical Number Systems is pointed out, as well as the relation to the Schur-Cohn region introduced in Chapter 2. Chapter 4, New algorithms and topological results: Two new algorithms which allow the characterization of SRS with finiteness property are presented here. The algorithms are then applied to settle two previously open questions on the topology of the set characterizing 2-dimensional SRS with finiteness property: It is shown that this set is disconnected and that the largest connected component has a non-trivial fundamental group. Chapter 5, Gaussian Shift Radix Systems and Pethõ's Loudspeaker: In this chapter SRS are generalized to complex numbers and Gaussian integers which leads to the definition of Gaussian Shift Radix Systems (GSRS). A conjecture on the set characterizing all 1-dimensional GSRS with finiteness property is formulated and proved in substantial parts. Furthermore it is shown that this set - known as Pethõ's Loudspeaker - has a critical point and possesses a kind of self-similarity that is revealed by the GSRS analogue of one of the algorithms introduced in Chapter 4. Chapter 6, Shift Radix Systems over imaginary quadratic Euclidean domains: Another generalization of SRS to imaginary quadratic Euclidean domains is considered here. The surprising observation that two of the five domains seem to have critical points while one has only weakly critical points and the remaining two have neither, is proved in parts. Some of the results of this thesis have appeared or will appear in: Characterization algorithms for shift radix systems with finiteness property, Weitzer M., 2015, Int. J. Number Theory, 11(1) On the characterization of Pethõ's Loudspeaker, Weitzer M., 2015, Publ. Math. Debrecen. To appear. A number theoretic problem on the distribution of polynomials with bounded roots, Kirschenhofer P. and Weitzer M., 2015, Integers, 15(#A10)
Translated title of the contributionShift Radix Systeme und deren Verallgemeinerungen
Original languageEnglish
QualificationDr.mont.
Supervisors/Advisors
  • Kirschenhofer, Peter, Assessor A (internal)
  • Pethõ, Attila, Assessor B (external), External person
  • Tichy, Robert, Assessor B (external), External person
Publication statusPublished - 2015

Bibliographical note

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Keywords

  • shift radix systems
  • numeration systems
  • almost linear recurrences
  • discrete dynamical systems
  • finite and periodic orbits
  • Schur-Cohn region
  • Selberg and Aomoto integrals

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