Projectional entropy and the electrical wire shift

Abstract

In this paper we present an extendible, block gluing $\mathbb Z^3$ shift of finite type $W^{\text{el}}$ in which the topological entropy equals the $L$-projectional entropy for a two-dimensional sublattice $L:=\mathbb Z \vec{e}_1+\mathbb Z\vec{e}_2\subsetneq\mathbb Z^3$, even so $W^{\text{el}}$ is not a full $\mathbb Z$ extension of $W^{\text{el}}_L$. In particular this example shows that Theorem 4.1 of [3] does not generalize to $r$-dimensional sublattices $L$ for $r>1$. Nevertheless we are able to reprove and extend the result about one-dimensional sublattices for general (non-SFT) $\mathbb Z^d$ shifts under the same mixing assumption as in [3] and by posing a stronger mixing condition we also obtain the corresponding statement for higher-dimensional sublattices.