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.