Introduction to hierarchical tiling dynamical systems

TL;DR

This paper surveys hierarchical tiling dynamical systems, comparing various types and focusing on supertile construction methods, their dynamical analysis, and spectral properties in the context of aperiodic order.

Contribution

It provides a comprehensive comparison of different hierarchical tiling systems and discusses their dynamical and spectral analysis methods, highlighting recent developments.

Findings

Comparison of symbolic and tiling systems

Analysis of spectral properties of supertile systems

Discussion of dynamical techniques for aperiodic order

Abstract

This paper is about the tiling dynamical systems approach to the study of aperiodic order. We compare and contrast four related types of systems: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional $Z^d$-systems, and multidimensional tiling systems. Aperiodically ordered structures are often hierarchical in nature, and there are a number of different yet related ways to define them. We will focus on what we are calling "supertile construction methods": symbolic substitution in one and many dimensions, S-adic sequences, self-similar and pseudo-self-similar tilings, and fusion rules. The techniques of dynamical analysis of these systems are discussed and a number of results are surveyed. We conclude with a discussion of the spectral theory of supertile systems from both the dynamical and diffraction perspectives.