Describe attractors of a finite family of contraction mappings

Question

The question is to describe the attractor of iterated function system $\mathcal{F}=\{R^2,f_1,f_2\},$ where $f_1,f_2$ are the two affine transformations$\begin{bmatrix} 0 & 0.8\\ -0.5&0 \\\end{bmatrix}.\begin{bmatrix} x\\ y \\\end{bmatrix}+\begin{bmatrix} 0\\ 0.5\\\end{bmatrix}$;$\begin{bmatrix} 0.7 & 0\\ 0&0.6 \\\end{bmatrix}.\begin{bmatrix} x\\ y \\\end{bmatrix}+\begin{bmatrix} 0.6\\ 0.4\\\end{bmatrix}$. Here $f_1$ and $f_2$ are contraction mappings , hence there must be an unique compact set $B \subset R^2$ which satisfies $ B=f_1(B)\bigcup f_2(B) $ . My question is , how to find this $B?$ I've tried to use the fact that $B=\lim_{n\rightarrow\infty}(f_1\bigcup f_2)^n(X)$ for any compact set $X\subset R^2$. However, the calculation is tedious and I couldn't find the result. Is there any convenient way to find the attractor?

Answer 1

The attractor is the union of the red and blue portions shown here:

The red portion is $f_1(B)$ and the blue portion is $f_2(B)$. As suggested in the comments, I generated the image on a computer. There is still quite a lot that can be determined analytically, however. The attractor is totally disconnected, for example. The convex hull can also be determined, as illustrated in the diagram. The extreme point in the upper right is the fixed point of $f_2$, namely $x_0=(2,1)$, while the other extreme points are $f_1(x_0)$, $f_1^2(x_0)$, $f_1^3(x_0)$, and $f_2^k(f_1^3(x_0))$, where $k$ ranges from $1$ to $\infty$.