Rauzy Fractal of the Smallest Substitution Associated with the Smallest Pisot Number

ABSTRACT

Up to words reversal and relabeling, there exists a unique substitution associated with the smallest Pisot number with a minimal number of letters. This is the substitution s: 1↦2, 2↦3, 3↦12. We study the Rauzy fractal of this substitution and show that it is the union of a countable number of Hokkaido tiles and a fractal of dimension strictly less than 2 which is completely explicit. We complete the picture by showing that these Hokkaido tiles are arranged in three different manners to form tiles which are all pairwise disjoint. We also give an efficient algorithm to draw a zoom on a Rauzy fractal. And we show that the symbolic system of the substitution s is measurably isomorphic to a nice domain exchange with $\text{four}$ pieces. The tools used in this article, using regular languages, are very general and can be easily adapted to study Rauzy fractals of any substitution associated with a Pisot number, and other fractals associated with algebraic numbers without conjugate of modulus one.

Keywords:

• Rauzy fractals
• substitutions
• Hokkaido
• Pisot number

2010 AMS SUBJECT CLASSIFICATION:

• 28A80
• 37B10
• 37F20
• 68R15

Additional information

Funding

Aix-Marseille Université