Rauzy fractal

Last updated April 10, 2023

In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution

It was studied in 1981 by Gérard Rauzy,^[1] with the idea of generalizing the dynamic properties of the Fibonacci morphism. That fractal set can be generalized to other maps over a 3-letter alphabet, generating other fractal sets with interesting properties, such as periodic tiling of the plane and self-similarity in three homothetic parts.

Definitions

Tribonacci word

The infinite tribonacci word is a word constructed by iteratively applying the Tribonacci or Rauzy map : $s(1)=12$ , $s(2)=13$ , $s(3)=1$ .^[2]^[3] It is an example of a morphic word. Starting from 1, the Tribonacci words are:^[4]

$t_{0}=1$
$t_{1}=12$
$t_{2}=1213$
$t_{3}=1213121$
$t_{4}=1213121121312$

We can show that, for $n>2$ , $t_{n}=t_{n-1}t_{n-2}t_{n-3}$ ; hence the name "Tribonacci".

Fractal construction

Consider, now, the space $R^{3}$ with cartesian coordinates (x,y,z). The Rauzy fractal is constructed this way:^[5]

1) Interpret the sequence of letters of the infinite Tribonacci word as a sequence of unitary vectors of the space, with the following rules (1 = direction x, 2 = direction y, 3 = direction z).

2) Then, build a "stair" by tracing the points reached by this sequence of vectors (see figure). For example, the first points are:

$1\Rightarrow (1,0,0)$
$2\Rightarrow (1,1,0)$
$1\Rightarrow (2,1,0)$
$3\Rightarrow (2,1,1)$
$1\Rightarrow (3,1,1)$

etc...Every point can be colored according to the corresponding letter, to stress the self-similarity property.

3) Then, project those points on the contracting plane (plane orthogonal to the main direction of propagation of the points, none of those projected points escape to infinity).

Properties

Can be tiled by three copies of itself, with area reduced by factors $k$ , $k^{2}$ and $k^{3}$ with $k$ solution of $k^{3}+k^{2}+k-1=0$ : $\scriptstyle {k={\frac {1}{3}}(-1-{\frac {2}{\sqrt[{3}]{17+3{\sqrt {33}}}}}+{\sqrt[{3}]{17+3{\sqrt {33}}}})=0.54368901269207636}$ .
Stable under exchanging pieces. We can obtain the same set by exchanging the place of the pieces.
Connected and simply connected. Has no hole.
Tiles the plane periodically, by translation.
The matrix of the Tribonacci map has $x^{3}-x^{2}-x-1$ as its characteristic polynomial. Its eigenvalues are a real number $\beta =1.8392$ , called the Tribonacci constant, a Pisot number, and two complex conjugates $\alpha$ and ${\bar {\alpha }}$ with $\alpha {\bar {\alpha }}=1/\beta$ .
Its boundary is fractal, and the Hausdorff dimension of this boundary equals 1.0933, the solution of $2|\alpha |^{3s}+|\alpha |^{4s}=1$ .^[6]

Variants and generalization

For any unimodular substitution of Pisot type, which verifies a coincidence condition (apparently always verified), one can construct a similar set called "Rauzy fractal of the map". They all display self-similarity and generate, for the examples below, a periodic tiling of the plane.

s(1)=12, s(2)=31, s(3)=1
s(1)=12, s(2)=23, s(3)=312
s(1)=123, s(2)=1, s(3)=31
s(1)=123, s(2)=1, s(3)=1132

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References

↑ Rauzy, Gérard (1982). "Nombres algébriques et substitutions" (PDF). Bull. Soc. Math. Fr. (in French). 110: 147–178. Zbl 0522.10032.
↑ Lothaire (2005) p.525
↑ Pytheas Fogg (2002) p.232
↑ Lothaire (2005) p.546
↑ Pytheas Fogg (2002) p.233
↑ Messaoudi, Ali (2000). "Frontière du fractal de Rauzy et système de numération complexe. (Boundary of the Rauzy fractal and complex numeration system)" (PDF). Acta Arith. (in French). 95 (3): 195–224. Zbl 0968.28005.

Arnoux, Pierre; Harriss, Edmund (August 2014). "WHAT IS... a Rauzy Fractal?". Notices of the American Mathematical Society. 61 (7): 768–770. doi: 10.1090/noti1144 .
Berthé, Valérie; Siegel, Anne; Thuswaldner, Jörg (2010). "Substitutions, Rauzy fractals and tilings". In Berthé, Valérie; Rigo, Michel (eds.). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge: Cambridge University Press. pp. 248–323. ISBN 978-0-521-51597-9. Zbl 1247.37015.
Lothaire, M. (2005). Applied combinatorics on words . Encyclopedia of Mathematics and its Applications. Vol. 105. Cambridge University Press. ISBN 978-0-521-84802-2. MR 2165687. Zbl 1133.68067.
Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.

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[1] Rauzy, Gérard (1982). "Nombres algébriques et substitutions" (PDF). Bull. Soc. Math. Fr. (in French). 110: 147–178. Zbl 0522.10032.

[ACOW525-2] Lothaire (2005) p.525

[PF232-3] Pytheas Fogg (2002) p.232

[ACOW546-4] Lothaire (2005) p.546

[PF233-5] Pytheas Fogg (2002) p.233

[6] Messaoudi, Ali (2000). "Frontière du fractal de Rauzy et système de numération complexe. (Boundary of the Rauzy fractal and complex numeration system)" (PDF). Acta Arith. (in French). 95 (3): 195–224. Zbl 0968.28005.

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