Vertices of self-similar tiles

Abstract

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 $n\ge 1$.