Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

Abstract

<p style='text-indent:20px;'>In this text I study the asymptotics of the complexity function of <i>minimal</i> 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 [<xref ref-type="bibr" rid="b12">12</xref>] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the <inline-formula><tex-math id="M1">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M2">\begin{document}$ [0, d] $\end{document}</tex-math></inline-formula>. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all <inline-formula><tex-math id="M3">\begin{document}$ d \ge 2 $\end{document}</tex-math></inline-formula> the possible values for entropy dimensions of <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional SFT are the <inline-formula><tex-math id="M5">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M6">\begin{document}$ [0, d-1] $\end{document}</tex-math></inline-formula>. In the present text I prove formally this result for <inline-formula><tex-math id="M7">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case <inline-formula><tex-math id="M8">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> requires some substantial changes to be made in order to adapt the construction that are not discussed here.</p>