The relationship between word complexity and computational complexity in subshifts

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