On the boundaries of self-similar tiles in ℝ¹

He, Xing-Gang

doi:10.1090/S0002-9939-06-08643-6

On the boundaries of self-similar tiles in $\mathbb {R}^1$
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by Xing-Gang He

Proc. Amer. Math. Soc. 134 (2006), 3163-3170

DOI: https://doi.org/10.1090/S0002-9939-06-08643-6

Published electronically: June 5, 2006

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Abstract:

The aim of this note is to study the construction of the boundary of a self-similar tile, which is generated by an iterated function system $\{\phi _i(x)=\frac {1}{N} (x+d_i)\}_{i=1}^N$. We will show that the boundary has complicated structure (no simple points) in general; however, it is a regular fractal set.

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