Renewal systems and their generating sets

Abstract

Renewal systems are symbolic dynamical systems whose points are bi-infinite concatenations of finite words. The purpose of this thesis is to study dynamical properties of renewal systems and combinatorial structures of their generating sets. First, we present three equivalent conditions for a generating set of a renewal system to generate a maximal monoid in the language of the system. We show that if a code generates a shift of finite type and satisfies those conditions, then it is cyclic. Sufficient conditions are given when the converse holds. Also we find a condition under which low step renewal systems of finite type. Second, the set of entropies of shifts of finite type is proven to be the same as the set of entropies of almost cyclic renewal systems. The period of a finite set of words is the greatest common divisor of the lengths of the words. We show that the period of a renewal system as a shift space and the minimum of the periods of its generating sets coincide when the system is of finite type or mixing. In the last, the zeta functions of uniquely decipherable renewal systems and almost cyclic renewal systems are computed based on the graph presentations for their generating sets. They are used to find combinatorial properties of the generating sets of renewal systems. More specifically, a pure code generating a shift of finite type is cyclic.