Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type
Let \(\mathcal{A}\) be an alphabet, that is any finite set and \(d\) a positive integer. The shift of dimension \(d\) over \(\mathcal{A}\) is the dynamical system obtained by the action of \(\mathbb{Z}^d\) on \(\mathcal{A}^{\mathbb{Z}^d}\) by shifting the indices. By Tychonoff theorem, \(\mathcal{A}^{\mathbb{Z}^d}\)is compact for the product topology. A subshift is merely a \(\mathbb{Z}^d\)-invariant closed subset. Roughly speaking, a subshift \(X\) is of finite type if the belonging of \(x\in\mathcal{A}^{\mathbb{Z}^d}\) to \(X\) can be decided by finitely patterns of finite size. A subshift is minimal if there is no non-empty invariant closed subset. It is known that minimal multidimensional subhifts of finite type have entropy zero. Here the author uses the entropy dimension that is a topological asymptotic invariant of the subshift. It was known thanks to results by \textit{M. Hochman} and \textit{T. Meyerovitch} [Ann. Math. (2) 171, No. 3, 2011--2038 (2010; Zbl 1192.37022)] that entropy dimensions are \(\pi_1\)-computable numbers. The main result of this paper is the fact that minimal tridimensional subshifts of finite type have entropy dimensions that are \(\Delta_2\)-computable numbers. Recall that a real number \(x\) is \(\Delta_2\)-computable if there is a Turing machine that produces a sequence of rational numbers \((r_n)\) such that \(x=\lim_n r_n\). The author explains that his ideas allow to get the same result in higher dimension (even if no formal proof is provided). In dimension two the situation is different and is not treated here. Without the finite-type assumption, in dimension two, it is known that any non-negative real number at most 2 can be realized as entropy dimension of a subshift. For the main result, the construction of the desired subshift is illustrated by a large number of pictures. The author states that a longer version of the paper with proofs of well-known facts is available but there is no reference to it and it seems not available neither in the webpage of the author nor at \url{https://arxiv.org/}.
scientific article; zbMATH DE number 7463736
scientific article