The relationship between word complexity and computational complexity in subshifts

Abstract

We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing spectrum can be realized via a subshift of linear complexity if and only if it consists of the union of a finite set and a finite number of cones, that a Turing spectrum can be realized via a subshift of exponential complexity (i.e. positive entropy) if and only if it contains a cone, and that every Turing spectrum which either contains degree 0 or is a union of cones is realizable by subshifts with a wide range of 'intermediate' complexity growth rates between linear and exponential.

Computability of topological pressure on compact shift spaces beyond finite type*

“…Much of what is presented here is already known, but we include this short section for completeness. For additional details, see, for example [3,28,33,34,50], and the references therein.…”