Vertices of Self-Similar Tiles

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 ℝ² 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 Lévy 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 Lévy dragon for all n ≥ 1.