Projectional entropy and the electrical wire shift

TL;DR

This paper constructs a specific three-dimensional shift of finite type demonstrating that certain entropy properties do not extend to higher-dimensional sublattices, and extends related results to more general shifts.

Contribution

It provides a counterexample for higher-dimensional sublattices and extends entropy results to broader classes of shifts under mixing conditions.

Findings

Counterexample for $r$-dimensional sublattices with $r>1$

Reproves and extends entropy results for non-SFT shifts

Shows stronger mixing conditions yield higher-dimensional sublattice results

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.