Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

In this text I study the asymptotics of the complexity function of minimal multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [ 12 ] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the \begin{document}$ \Delta_2 $\end{document} -computable numbers in \begin{document}$ [0, d] $\end{document} . The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all \begin{document}$ d \ge 2 $\end{document} the possible values for entropy dimensions of \begin{document}$ d $\end{document} -dimensional SFT are the \begin{document}$ \Delta_2 $\end{document} -computable numbers in \begin{document}$ [0, d-1] $\end{document} . In the present text I prove formally this result for \begin{document}$ d = 3 $\end{document} . Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case \begin{document}$ d = 2 $\end{document} requires some substantial changes to be made in order to adapt the construction that are not discussed here.