  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.
