Fractals and Kronecker product

A few years ago, I went to listen to Roger Nelsen who was giving a talk about copulas with fractal support. Roger is amazing when he gives a talk (I am also a huge fan of his books, and articles), and I really wanted to play with that concept (that he did publish later on, with Gregory Fredricks and José Antonio Rodriguez-Lallena). I did mention that idea in a paper, writen with Alessandro Juri, just to mention some cases where deriving fixed point theorems is not that simple (since the limit may not exist).

The idea in the initial article was to start with something quite simple, a the so-called transformation matrix, e.g.

https://latex.codecogs.com/gif.latex?T=\frac{1}{8}\left(\begin{matrix}1&%200%20&%201%20\\%200%20&%204%20&%200%20\\%201%20&%200&1\end{matrix}\right)
Here, in all areas with mass, we spread it uniformly (say), i.e. the support of https://latex.codecogs.com/gif.latex?T(C^\perp) is the one below, i.e. https://latex.codecogs.com/gif.latex?1/8th of the mass is located in each corner, and https://latex.codecogs.com/gif.latex?1/2 is in the center. So if we spread the mass to have a copula (with uniform margin,)we have to consider squares on intervals https://latex.codecogs.com/gif.latex?[0,1/4]https://latex.codecogs.com/gif.latex?[1/4,3/4] and https://latex.codecogs.com/gif.latex?[3/4,1],

Then the idea, then, is to consider https://latex.codecogs.com/gif.latex?T^2=\otimes^2T, where  https://latex.codecogs.com/gif.latex?\otimes^2T is the tensor product (also called Kronecker product) of https://latex.codecogs.com/gif.latex?T with itself. Here, the support of https://latex.codecogs.com/gif.latex?T^2(C^\perp) is

Then, consider https://latex.codecogs.com/gif.latex?T^3=\otimes^3T, where https://latex.codecogs.com/gif.latex?\otimes^3T is the tensor product of https://latex.codecogs.com/gif.latex?T with itself, three times. And the support of https://latex.codecogs.com/gif.latex?T^3(C^\perp) is

Etc. Here, it is computationally extremely simple to do it, using this Kronecker product. Recall that if https://latex.codecogs.com/gif.latex?%20%20%20%20%20\mathbf{A}=(a_{i,j}), then

https://latex.codecogs.com/gif.latex?%20%20%20%20%20\mathbf{A}\otimes\mathbf{B}%20=%20\begin{pmatrix}%20a_{11}%20\mathbf{B}%20&%20\cdots%20&%20a_{1n}\mathbf{B}%20\\%20\vdots%20&%20\ddots%20&%20\vdots%20\\%20a_{m1}%20\mathbf{B}%20&%20\cdots%20&%20a_{mn}%20\mathbf{B}%20\end{pmatrix}

So, we need a transformation matrix: consider the following https://latex.codecogs.com/gif.latex?4\times4 matrix,

> k=4
> M=matrix(c(1,0,0,1,
+            0,1,1,0,
+            0,1,1,0,
+            1,0,0,1),k,k)
> M
[,1] [,2] [,3] [,4]
[1,]    1    0    0    1
[2,]    0    1    1    0
[3,]    0    1    1    0
[4,]    1    0    0    1

Once we have it, we just consider the Kronecker product of this matrix with itself, which yields a https://latex.codecogs.com/gif.latex?4^2\times4^2 matrix,

> N=kronecker(M,M)
> N[,1:4]
[,1]  [,2] [,3] [,4]
[1,]     1    0    0    1
[2,]     0    1    1    0
[3,]     0    1    1    0
[4,]     1    0    0    1
[5,]     0    0    0    0
[6,]     0    0    0    0
[7,]     0    0    0    0
[8,]     0    0    0    0
[9,]     0    0    0    0
[10,]    0    0    0    0
[11,]    0    0    0    0
[12,]    0    0    0    0
[13,]    1    0    0    1
[14,]    0    1    1    0
[15,]    0    1    1    0
[16,]    1    0    0    1

And then, we continue,

> for(s in 1:3){N=kronecker(N,M)}

After only a couple of loops, we have a https://latex.codecogs.com/gif.latex?4^5\times4^5 matrix. And we can plot it simply to visualize the support,

> image(N,col=c("white","blue"))

As we zoom in, we can visualize this fractal property,


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