TEL-AVIV UNIVERSITY RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCES SCHOOL OF MATHEMATICAL SCIENCES Tail Invariant Measures of The Dyck-Shift and Non-Sofic Systems

Among the most familiar systems in symbolic dynamics are the subshifts of finite type, or SFT’s for short. SFT’s are relatively easy to analyze, and have many pleasant properties such as intrinsic ergodicity and unique ergodicity with respect to the tail. A larger class, which has the desirable property of being closed under factors, is that of sofic systems. This class of systems retains many of the properties of SFT’s, and has several convenient characterizations. In the past few decades there have been attempts to find generalizations of sofic systems. In chapter 2 we study some work attempting to generalize sofic systems. Blanchard’s concept of ”coded systems”, and Buzzi’s concept of subshifts of ”Quasi-Finite type” are reviewed. We also explore the class of β-shifts in the context of non-sofic subshifts. We discuss at length a class of non-sofic systems, known as the Dyck systems. We show that the one-sided Dyck shift has a unique tail invariant tail-invariant topologically σ-finite measure (up to scaling). This invariant measure of the one sided Dyck turns out to be a shift-invariant probability. Furthermore, it is one of the two ergodic probabilities obtaining maximal entropy. We also prove that the two sided Dyck has a double-tail invariant probability, which is also shift invariant, with entropy strictly less than the topological entropy.