Journal of the Mathematical Society of Japan

Boundary parametrization of self-affine tiles

Shigeki AKIYAMA and Benoît LORIDANT

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Abstract

A standard way to parametrize the boundary of a connected fractal tile T is proposed. The parametrization is Hölder continuous from R/Z to ∂T and fixed points of ∂T have algebraic preimages. A class of planar tiles is studied in detail as sample cases and a relation with the recurrent set method by Dekking is discussed. When the tile T is a topological disk, this parametrization is a bi-Hölder homeomorphism.

Article information

Source
J. Math. Soc. Japan, Volume 63, Number 2 (2011), 525-579.

Dates
First available in Project Euclid: 25 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1303737797

Digital Object Identifier
doi:10.2969/jmsj/06320525

Mathematical Reviews number (MathSciNet)
MR2793110

Zentralblatt MATH identifier
1209.28004

Subjects
Primary: 28A80: Fractals [See also 37Fxx] 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20] 68Q70: Algebraic theory of languages and automata [See also 18B20, 20M35]
Secondary: 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 28A78: Hausdorff and packing measures 37F20: Combinatorics and topology 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 54D05: Connected and locally connected spaces (general aspects)

Keywords
self-affine tile graph directed set Hausdorff measure Büchi automata

Citation

AKIYAMA, Shigeki; LORIDANT, Benoît. Boundary parametrization of self-affine tiles. J. Math. Soc. Japan 63 (2011), no. 2, 525--579. doi:10.2969/jmsj/06320525. https://projecteuclid.org/euclid.jmsj/1303737797


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