Proximality and pure point spectrum for tiling dynamical systems

TL;DR

This paper explores the relationship between proximality and spectral properties in tiling dynamical systems, establishing conditions under which proximality is closed and linking it to eigenfunctions and minimal rank.

Contribution

It provides new equivalences connecting proximality, minimal rank, and eigenfunctions in tiling dynamical systems under specific hypotheses.

Findings

Proximality is topologically closed if and only if minimal rank is one.

The set of fiber distal points has full measure under the hypotheses.

Meyer property is key to relating proximality to strong proximality.

Abstract

We investigate the role of the proximality relation for tiling dynamical systems. Under two hypotheses, namely that the minimal rank is finite and the set of fiber distal points has full measure we show that the following conditions are equivalent: (i) proximality is topologically closed, (ii) the minimal rank is one, (iii) the continuous eigenfunctions of the translation action span the L^2-functions over the tiling space. We apply our findings to model sets and to Meyer substitution tilings. It turns out that the Meyer property is crucial for our analysis as it allows us to replace proximality by the a priori stronger notion of strong proximality.