Homomorphisms between multidimensional constant-shape substitutions

TL;DR

This paper investigates homomorphisms between multidimensional constant-shape substitutions, proving that measurable factors are continuous and establishing strong restrictions on the normalizer group, including invertibility of endomorphisms.

Contribution

It introduces new results on the continuity of homomorphisms and characterizes the structure of the normalizer group for these substitutions.

Findings

Measurable factor maps are continuous.

Endomorphisms are invertible.

Normalizer group is virtually generated by shift actions.

Abstract

We study a class of $\Z^{d}$-substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of $\Z^{d}$. We prove that any measurable factor map and even any homomorphism associated to a matrix commuting with the expansion matrix, induces a continuous one. We also get strong restrictions on the normalizer group, proving that any endomorphism is invertible, the normalizer group is virtually generated by the shift action and the quotient of the normalizer group by the automorphisms is restricted by the digit tile of the substitution.