Factoring onto $\mathbb{Z}^d$ subshifts with the finite extension property

TL;DR

This paper introduces the finite extension property for d-dimensional subshifts, proves its invariance under topological conjugacy, and shows that certain subshifts factor onto lower entropy subshifts, extending previous results.

Contribution

It generalizes the topological strong spatial mixing condition and extends factorization results to a broader class of subshifts without needing a safe symbol.

Findings

Finite extension property is invariant under topological conjugacy.

Every d-dimensional block gluing subshift factors onto lower entropy subshifts with the property.

Extension of previous theorems to broader classes of subshifts.

Abstract

We define the finite extension property for $d$-dimensional subshifts, which generalizes the topological strong spatial mixing condition defined by Brice\~no (2016), and we prove that this property is invariant under topological conjugacy. Moreover, we prove that for every $d$, every $d$-dimensional block gluing subshift factors onto every $d$-dimensional subshift which has strictly lower entropy, a fixed point, and the finite extension property. This result extends a theorem from Boyle, Pavlov, and Schraudner (2010), which requires that the factor contain a safe symbol.