Fractal tiles induced by tent maps

TL;DR

The article classifies fractal tent-tiles arising from tent maps tied to special Pisot numbers, showing these tiles have positive Lebesgue measure and fractal boundaries whose Hausdorff dimensions are computable or tightly bounded. By relating tent-tiles to Rauzy fractals through GIFS and substitutions (including automorphisms of free groups), the authors derive explicit correspondences and tiling properties, including two lattice-tiling types and computations of self-replicating and lattice boundary graphs to certify tilings. The work provides a finite, structured catalog of planar and three-dimensional cases, with detailed dimension formulas and tiling conditions, supported by algorithmic computations (e.g., via Mathematica). This connects dynamical tent-map geometry with Rauzy fractal theory, enriching both the fractal tiling literature and the understanding of special Pisot parameters in tiling dynamics, and it offers concrete methods to determine tiling behavior and boundary geometry for each case.

Abstract

In the present article, we deal with geometrical objects induced by the tent maps associated with special Pisot numbers that we call tent-tiles. They are compact subsets of the one-, two-, or three-dimensional Euclidean space, depending on the particular special Pisot number. Most of the tent-tiles have a fractal shape and we study the Hausdorff dimension of their boundary. Furthermore, we are concerned with tilings induced by tent-tiles. It turns out that tent-tiles give rise to two types of lattice tilings. In order to obtain these results we establish and exploit connections between tent-tiles and Rauzy fractals induced by substitutions and automorphisms of the free group.

Figures (9)

• Figure 1: There are six planar tent-tiles. They all have a fractal shape.
• Figure 2: Two types of lattice tilings induced by tent-tiles. On the left hand side the tile itself produces the tiling. On the right hand side the tiling is generated by the tile and its reflection.
• Figure 3: The tent-tiles are given by the invariant set of the GIFS determined by the depicted graph.
• Figure 4: Prefix graph of the substitution $\zeta_p$.
• Figure 5: Prefix graph of the substitution $\theta_q$.

Theorems & Definitions (52)

• Theorem 1.1
• Proposition 2.1: cf. Bandt.1989b, Mauldin-Williams:88
• Definition 2.2
• Theorem 2.3: cf. Lagarias-Porta-Stolarsky:94
• proof
• Remark 3.1
• Proposition 3.2: Sirvent-Wang.2002
• Definition 3.3: Coincidence condition
• Lemma 3.4
• proof