An Approximation Theorem for Maps Between Tiling Spaces

TL;DR

This paper proves that continuous maps between translationally finite tiling spaces can be approximated by local maps, and homotopies between local maps can also be made local, advancing understanding of tiling space mappings.

Contribution

It introduces an approximation theorem for maps between tiling spaces, showing all continuous maps can be approximated by local ones and homotopies can be localized.

Findings

Continuous maps can be approximated by local maps.

Homotopies between local maps can be chosen to be local.

Advances the theoretical understanding of tiling space mappings.

Abstract

We show that every continuous map from one translationally finite tiling space to another can be approximated by a local map. If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.