Some Recent Developments of Self-Affine Tiles

Abstract

A self-affine set \(T:= T(A,\mathcal{D})\) is the attractor of an affine pair \((A,\mathcal{D})\), where A is an expanding matrix on \(\mathbb{R}^{s}\) with integral entries, and \(\mathcal{D} \subset \mathbb{Z}^{s}\) is a finite set; T is called a self-affine tile if it is also a tile, and call such \(\mathcal{D}\) a tile digit set. In this survey, we review some recent developments on the structure and characterizations of the tile digit sets \(\mathcal{D}\) for a given A. We also discuss the celebrated Fuglede’s spectral set problem on the self-affine tiles.