Shifts of finite type associated to an integer matrix

Abstract: Shifts of finite type are one of the fundamental objects in the field of symbolic dynamics. These are the spaces of one-sided or two-sided sequences over a finite set of symbols where certain finitely many words are "forbidden". The shift spaces exhibit a natural association with $0-1$ matrices. In this talk we will first discuss some necessary preliminaries related to the one-sided shifts of finite type associated to a non-negative integer matrix. Words which correspond to the matrix entries greater than $1$ are thought to have multiplicity and thus are called "repeated words". Now, for any given collection $\mathcal{F}$ of forbidden words and $\mathcal{R}$ of repeated words, we define two notions: multiplicity of a word and generalized language. We define the shift determined by $\mathcal{F}$ and $\mathcal{R}$, and obtain necessary and sufficient conditions for when the language of this shift is precisely the generalized language. Finally, we compute the entropy of this shift using the generalized language and study some properties of Markov measures on the shift.

This talk is based on a joint work with Agarwal N. and Haritha C.