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

TL;DR

This paper proves that finite type property holds for self-similar sets with complex Pisot expansion factors, broadening the class of sets with well-understood separation properties and natural textures.

Contribution

It establishes that finite type property always holds for complex Pisot factors in the algebraic number field, extending previous real-case results.

Findings

Finite type property holds for complex Pisot factors.

Extension of previous real-case results to complex numbers.

Provides numerous examples of separated self-similar sets.

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.