On some aspects of cocyclic subshifts, languages, and automata

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.