Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Question

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group generated by the permutation $$ a: \ (m,n) \ \mapsto \ (m-n,m) $$ of order $6$ and the involutions $$ b: \ (m,n) \ \mapsto \ \begin{cases} (m,2n+1) & \text{if} \ n \equiv 0(\text{mod} \ 2), \\ (m,(n-1)/2) & \text{if} \ n \equiv 1(\text{mod} \ 4), \\ (m,n) & \text{if} \ n \equiv 3(\text{mod} \ 4) \\ \end{cases} $$ and $$ c: \ (m,n) \ \mapsto \ \begin{cases} (m,2n+3) & \text{if} \ n \equiv 0(\text{mod} \ 2), \\ (m,(n-3)/2) & \text{if} \ n \equiv 3(\text{mod} \ 4), \\ (m,n) & \text{if} \ n \equiv 1(\text{mod} \ 4). \\ \end{cases} $$ Drawing spheres of radius $r$ about $(0,0)$ for "large enough" $r$ reveals fractal-like structures.

Added on July 28, 2014: A video showing more pictures is now available on YouTube here. The video starts with a sequence showing entire spheres of small radii, i.e. from $r = 8$ to $r = 24$, and continues with pictures showing smaller parts of spheres of larger radii up to $r = 45$. Monochrome pictures show only one sphere, respectively, a part thereof; colored pictures show multiple spheres in different colors.

Added on March 16, 2014: Pictures of the spheres of radius $30$ and $36$ can be downloaded here:

Radius 30 (3487 x 3079 pixels, 111KB), Radius 36 (10375 x 9103 pixels, 693KB).

Sample snippets of the large -- about $200$ megapixels at $r = 38$ to about $3$ gigapixels at $r = 45$ -- pictures are (black pixel = belongs to sphere, white pixel = doesn't belong to sphere):

The images above show parts of the spheres of radius $38$, $40$ and $45$.

Question: How can the observed patterns be explained?

Remark 1: The cardinalities of the spheres of radii $r = 0, \dots, 45$ about $(0,0)$ are

1, 2, 4, 8, 14, 26, 39, 68, 114, 188, 289, 404, 560, 827, 1341, 2052, 3158, 4540, 
6091, 8630, 12241, 17739, 27727, 41846, 61234, 86647, 117806, 163795, 233939, 340659, 
523862, 768739, 1110855, 1569204, 2148377, 2994661, 4287462, 6195498, 9389566, 
13568954, 19542862, 27619364, 38048372, 53304607, 76433012, 109839303.

The entire sphere of radius $20$ looks as follows (the overall shape of the larger spheres is roughly similar):

Added: At a scale of about $1:100$, the entire sphere of radius $45$ looks as follows:

Remark 2: In the notation of this and this question, $b$ and $c$ induce on the second coordinate the class transpositions $\tau_{0(2),1(4)}$ and $\tau_{0(2),3(4)}$, respectively. Further, in the notation of (1) and (2) we have $G < {\rm RCWA}(\mathbb{Z}^2)$.

Answer 1

This phenomenon can be understood as follows:

Let $\Gamma < Sym(\mathbb{Z}^{2})$ be a transitive hyperbolic group (and non virtually cyclic).

Through this faithful transitive action of $\Gamma$ on $\mathbb{Z}^{2}$ (and orbit of (0,0)), the Cayley graph of $\Gamma$ (Gromov hyperbolic geometry) is projected (in a manner not too degenerate) on $\mathbb{Z}^2$ (euclidean geometry).
Then the emergence of explosive and fractal-like structures on the spheres is not surprising.

So any (non virtually cyclic) hyperbolic group would generate beautiful fractal-like pictures.

Example: $\Gamma = \langle a,b \rangle < Sym(\mathbb{Z}^{2})$ such that

• $a: (m,n) \to (m+1,m+n)$
• $b: (m,n) \to (m+n,n+1)$

To be confirmed: $\Gamma$ acts transitively on $\mathbb{Z}^{2}$.
Remark: $\Gamma$ is neither free nor hyperbolic nor torsion-free (see the comments below).

The cardinalities of the spheres of radii $r=0,\dots,14$ about $(0,0)$ are :
$1,4,11,20,47,100,238,512,1124,2540,5569,12101,26208,56720,122600$

The entire sphere of radius $14$:

The entire ball of radius $14$ with a rainbow gradient according to the spheres:

Remark: I don't know if your group $G$ is hyperbolic, but we could have the same understanding for all the finitely generated groups of exponential growth.

Answer 2

This is just a partial answer, but it looks like your figures are not "pure" fractals, but a union of different fractals (with different fractal dimensions). If I am not mistaken, these are called multi-fractals.

It looks like part of your figure is the Dragon curve fractal.

This particular fractal is space-filling, so it would not surprise me if your set is actually everything, (unless I have misinterpreted something).

Also, your dynamics looks a lot like some two-dimensional version of the iterations of something similar to the Collatz conjecture, which give rise to a fractal behaviour. Look at the corresponding wikipedia page, there is an analytic continuation of the functions that are iterated, and I think something similar can be done in your case.

Ok, so you can make a continuous version of your map $b$: Let $s(n)=\frac12 \sin\frac{\pi n}{2} (1 + \sin \frac{\pi n}{2})$. Note that $s(n)$ is periodic, with period 4: $1,0,0,0,1,0,\dots$.

Now, we can define $b(m,n)$ as

$$ b(m,n)= \left(m, (2 n + 1) (s[n + 1] + s[n + 3]) + s[n + 2]n + s[n] (n - 1)/2 \right) $$ and this is now a continuous extension of your map, seen as a map on $\mathbb{R}^2$. You can do a similar thing with the other maps.

Now it might be easier to draw this, for example, this is exactly the kind of maps that the most common fractal drawing software works with. What you have is a Hutchinson operator, so check wikipedia again.