# Boundary parametrization of planar self-affine tiles with collinear digit set

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

2010-09-07

AUTHORS ABSTRACT

We consider a class of planar self-affine tiles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = M^{ - 1} \cup _{a \in \mathcal{D}} (T + a)$$\end{document} generated by an expanding integral matrix M and a collinear digit set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D}$$\end{document} as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M = \left( \begin{gathered} 0 - B \hfill \\ 1 - A \hfill \\ \end{gathered} \right),\mathcal{D} = \left\{ {\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} \right),...,\left( \begin{gathered} |B| - 1 \hfill \\ 0 \hfill \\ \end{gathered} \right)} \right\}$$\end{document}. We give a parametrization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{S}^1 \to \partial T$$\end{document} of the boundary of T with the following standard properties. It is Hölder continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on ∂T and have algebraic preimages. We derive a new proof that T is homeomorphic to a disk if and only if 2|A| ⩽ |B + 2|. More... »

PAGES

2173-2194

### References to SciGraph publications

• 1997-01. Integral self-affine tiles in ℝn part II: Lattice tilings in JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
• 1992. Number Systems and Fractal Geometry in PROBABILITY THEORY AND APPLICATIONS
• 2003. Neighbours of Self-affine Tiles in Lattice Tilings in FRACTALS IN GRAZ 2001
• 2001-01-01. Disk-Like Self-Affine Tiles in R2 in DISCRETE & COMPUTATIONAL GEOMETRY
• 1994-05. Self-Similar Lattice Tilings in JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
• ### Journal

TITLE

Science China Mathematics

ISSUE

9

VOLUME

53

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11425-010-4096-2

DOI

http://dx.doi.org/10.1007/s11425-010-4096-2

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1032513102

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