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2017 | OriginalPaper | Chapter

Some Recent Developments of Self-Affine Tiles

Authors : Chun-Kit Lai, Ka-Sing Lau

Published in: Recent Developments in Fractals and Related Fields

Publisher: Springer International Publishing

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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.

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Title: Some Recent Developments of Self-Affine Tiles
Authors: Chun-Kit Lai
Ka-Sing Lau
Publisher: Springer International Publishing
Book: Recent Developments in Fractals and Related Fields
Print ISBN: 978-3-319-57803-3

Electronic ISBN: 978-3-319-57805-7

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-57805-7_10

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