Dynamical Systems Seminar

A "random" shift of finite type over an alphabet A was defined by Kevin McGoff as follows: for any \(t\) in \([0,1]\) and positive integer \(n\), define a "random" set of words \(S\) which independently contains each possible word of length \(n\) with probability \(t\) (or omits it, with probability \(1-t)\). This defines a "random" shift of finite type \(X\) over \(A\) where \(S\) is the allowed set of \(n\)-letter words.

Fixing \(A\) and \(t\) and letting \(n\) approach infinity, he proved several interesting results about aspects of random shifts of finite type in the limit, such as the probability that \(X\) is empty and the topological entropy of \(X\).

I will present current joint work with Kevin in which we extend several of his results to \(\mathbb{Z}^d\) shifts of finite type. I'll also describe some of the fundamental differences between shifts of finite type in \(\mathbb{Z}\) and \(\mathbb{Z}^d\), and implications of our results in that context.

I'll assume some familiarity with one-dimensional shifts of finite type and basic dynamical notions such as periodic points and topological entropy, but most definitions will be briefly reviewed regardless.