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On Some Aspects of Cocyclic Subshifts, Languages, And ON SOME ASPECTS OF COCYCLIC SUBSHIFTS, LANGUAGES, AND AUTOMATA by David Bryant Buhanan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics MONTANA STATE UNIVERSITY Bozeman, Montana June, 2012 ⃝c COPYRIGHT by David Bryant Buhanan 2012 All Rights Reserved ii APPROVAL of a dissertation submitted by David Bryant Buhanan This dissertation has been read by each member of the dissertation committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to The Graduate School. Dr. Jaroslaw Kwapisz Approved for the Department of Mathematical Sciences Dr. Kenneth L. Bowers Approved for The Graduate School Dr. Carl Fox iii STATEMENT OF PERMISSION TO USE In presenting this dissertation in partial fulfillment of the requirements for a doc- toral degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. I further agree that copying of this dissertation is allowable only for scholarly purposes, consistent with \fair use" as pre- scribed in the U.S. Copyright Law. Requests for extensive copying or reproduction of this dissertation should be referred to ProQuest Information and Learning, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted \the exclusive right to reproduce and distribute my dissertation in and from microform along with the non-exclusive right to reproduce and distribute my abstract in any format in whole or in part." David Bryant Buhanan June, 2012 iv DEDICATION To Charlie, Willie, and Caixia. v ACKNOWLEDGEMENTS I am grateful to all of the people I have worked with at Montana State University. Special thanks are due to my committee members Lukas Geyer, Marcy Barge, Tomas Gedeon, and Mark Pernarowski for their questions and suggestions. Most impor- tantly, I am thankful to my thesis advisor Jarek Kwapisz for his help, support, and encouragement. Also thanks to my wife and children for their patience and support. vi TABLE OF CONTENTS 1. INTRODUCTION ........................................................................................1 2. DIOPHANTINE EQUATIONS AND COCYCLIC SUBSHIFTS ....................11 Preliminaries .............................................................................................. 11 Diophantine Equations ................................................................................ 13 Correspondence Theorem ............................................................................ 15 Undecidability ............................................................................................ 24 Incompleteness............................................................................................ 28 Strictly Cocyclic Subshifts and Diophantine Equations .................................. 29 Post Correspondence Problem and the Mortal Matrix Theorem ..................... 33 3. ALGEBRAIC PRELIMINARIES.................................................................37 Decompositions of Finite Dimensional Algebras ............................................ 37 The Radical and Semisimplicity ................................................................... 44 Semisimple Part of an Algebra..................................................................... 48 More on Principal Modules and Module Decomposition................................. 51 Miscellaneous Algebraic Results ................................................................... 53 4. MORSE DECOMPOSITION OF COCYCLIC SUBSHIFTS ..........................56 Wedderburn Decomposition of Cocyclic Subshifts ......................................... 56 Connections................................................................................................ 66 The Connecting Graph................................................................................ 71 Minimal Algebras and Transitivity of Connecting Diagrams........................... 75 Non-Unital Algebras ................................................................................... 79 SFT and Edge Shifts................................................................................... 84 Composition Series and Connecting Diagrams............................................... 86 Dynamical Disjointness ............................................................................... 87 Aperiodic Decomposition and Connecting Diagrams...................................... 90 Computation and Examples......................................................................... 95 5. LANGUAGES AND COCYCLIC SUBSHIFTS.............................................99 Background ................................................................................................ 99 Languages of Deterministic and Nondeterministic Cocycles.......................... 102 Cocyclic Automata and Languages............................................................. 106 Cocyclic Languages and Subshifts .............................................................. 118 REFERENCES CITED.................................................................................. 123 vii LIST OF FIGURES Figure Page 1.1 Simple Connecting Diagram...................................................................7 4.1 Connecting Diagram............................................................................ 74 4.2 Gabriel's Quiver .................................................................................. 75 4.3 Non-transitive Connecting Diagram...................................................... 75 4.4 Sofic System with Corresponding Non-Unital Algebra............................ 83 4.5 Graph for XΨ ...................................................................................... 84 4.6 Sofic system ........................................................................................ 89 4.7 Example ............................................................................................. 94 4.8 Connecting Graph - Simple Loop ......................................................... 96 4.9 Non-transitive Connecting Diagram ..................................................... 97 viii ABSTRACT In this thesis we examine the class of cocyclic subshifts. First, we give a method of modeling the zeros of Diophantine equations in the multiplication of matrices. This yields many examples, and a proof that there is no algorithm than can decide if two cocyclic subshifts are equal. Next, we use the theory of finite dimensional algebras to give an algebraic technique for finding connecting orbits between topologically transitive components of a given cocyclic subshift. This gives a very complete picture of the structure of the dynamical system associated to the cocyclic subshift. Last, we characterize the languages of cocyclic subshifts via cocyclic automata, in an analogous manner to the characterization of the languages of sofic subshifts which have languages accepted by finite automata. Our automaton definition is closely related to definitions in the field of quantum computation. 1 INTRODUCTION The long-term behavior of many dynamical systems can be well described by coding the system by a symbolic system, a procedure that goes back to at least Morse and Hedlund [30]. See [27, p.201] for an expositional introduction to coding. Coding involves partitioning the space into pieces and associating to each orbit the sequence of pieces visited by it. The collection of all of the possible sequences together with the shift map is a dynamical system. If the coding is done well, the symbolic system retains many properties of the original system, while being easier to manage. This technique was perhaps most famously employed in the proof that entropy is a complete metric invariant for toral automorphisms, see [1]. Symbolic systems were first introduced for coding, and after coding had proven useful, symbolic dynamics as an area in itself became interesting. Since symbolic sys- tems could be used to give information about other systems, it became important to understand the behavior of symbolic systems. Symbolic systems also arise naturally in the study of formal languages, an area in theoretical computer science. Although the theory of formal languages deals primarily with finite sequences of symbols (instead of infinite sequences), there are many parallels in the two theories and they benefit from each other. In this thesis we seek to understand the structure of symbolic systems that belong to the class of cocyclic subshifts. We start with the definition of the shift space. Let A be a finite alphabet, that is a finite set of symbols. Then the space AZ of infinite sequences with values in A and indexed by the integers Z is a compact metric space with 8 > < −k 2 : if x =6 y and k maximal so that x[−k;k] = y[−k;k] d(x; y) = > : 0 : if x = y 2 Z (Here x[i;j] := xixi+1 : : : xj for i ≤ j.) The full (two-sided) shift is the collection A Z together with the shift map f given on x = : : : x−2x−1:x0x1x2 ::: 2 A by f(x) = (yi)i N with yi = xi+1. We also look at A , where N are the natural numbers, with the shift map and refer to this system as the full (one-sided) shift. The one-sided shift space is also a metric space with the metric above modified to only check equality on x[1;k] and y[1;k]. A subset of the full shift which is closed and invariant under the shift map is known as a subshift. Before continuing with the discussion of specific classes of subshifts we provide the basic
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