Elementary fractal geometry. 3. Complex Pisot factors imply finite type

Abstract

Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive and finite Hausdorff measure for a weaker separation property which is always fulfilled for crystallographic data. Ngai and Wang gave more specific results for a finite type property (FT), and for algebraic data with a real Pisot expansion factor. We show how the algorithmic FT concept of Bandt and Mesing relates to the property of Ngai and Wang. Merits and limitations of the FT algorithm are discussed. Our main result says that FT is always true in the complex plane if the similarity mappings are given by a complex Pisot expansion factor $\lambda$ and algebraic integers in the number field generated by $\lambda .$ This extends the previous results and opens the door to huge classes of separated self-similar sets, with large complexity and an appearance of natural textures. Numerous examples are provided.