Illinois Journal of Mathematics

Vertices of self-similar tiles

Da-Wen Deng and Sze-Man Ngai

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The set $V_n$ of $n$-vertices of a tile $T$ in $\R^d$ is the common intersection of $T$ with at least $n$ of its neighbors in a tiling determined by $T$. Motivated by the recent interest in the topological structure as well as the associated canonical number systems of self-similar tiles, we study the structure of $V_n$ for general and strictly self-similar tiles. We show that if $T$ is a general self-similar tile in $\R^2$ whose interior consists of finitely many components, then any tile in any self-similar tiling generated by $T$ has a finite number of vertices. This work is also motivated by the efforts to understand the structure of the well-known L\'evy dragon. In the case $T$ is a strictly self-similar tile or multitile in $\R^d$, we describe a method to compute the Hausdorff and box dimensions of $V_n$. By applying this method, we obtain the dimensions of the set of $n$-vertices of the L\'evy dragon for all $n\ge 1$.

Article information

Illinois J. Math., Volume 49, Number 3 (2005), 857-872.

First available in Project Euclid: 13 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A80: Fractals [See also 37Fxx]
Secondary: 37B50: Multi-dimensional shifts of finite type, tiling dynamics 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]


Deng, Da-Wen; Ngai, Sze-Man. Vertices of self-similar tiles. Illinois J. Math. 49 (2005), no. 3, 857--872. doi:10.1215/ijm/1258138223.

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