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 $L:=\mathbb{Z}{\vec{e}}_1+\mathbb{Z}{\vec{e}}_2⊊{\mathbb{Z}}^3$, even so $W^{\text{el}}$ is not a full $\mathbb{Z}$ extension of $W_L^{\text{el}}$. 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.

12 pages, 3 figures