An iterated function system is defined as a finite set of contraction mappings, defined over a complete metric space $X$, and iteration is defined as sequential composition of these contraction mappings, where each mapping has some nonzero probability of being used at every iteration. Hutchinson (1981) proved that such iterated function systems (IFS) have invariant sets that iteration converges to, call this set $S \subset X$, so that for $n$ different contraction mappings defining our IFS, we have
$$
S = \bigcup_{i=1}^n f_i(S)
$$
I am wondering if such invariant sets $S$ are necessarily fractal if they are generated by an IFS. It is well know that many fractals can be generated by an IFS, but it is not clear if IFS's necessarily generate fractal invariant sets.
Are the invariant sets of all iterated function systems necessarily fractal
dynamical systemsfractalsiterated-function-system
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I can't give a complete answer, but I can make a few observations in the general direction of an answer. Note that most of this is based on memory from spending several hours looking at the behavior of these plots and observing how they relate to roots for individual polynomials. This is all "empirical math" (assuming that's even a thing, heh) but the patterns are pretty clear and while I doubt I have the mathematical background to make most of it rigorous I doubt it would be too difficult.
It's not limited to coefficients from $\{-1,1\}$; almost any collection of polynomials with coefficients chosen from a very limited set will produce similar patterns.
The character of the pattern is directly tied to the location on the complex plane, specifically the effect of multiplication. So I suspect that it catalogs the IFSs only insofar as the complex plane catalogs (some) affine transformations. Note that the Julia set formula, in contrast, lets the transformation vary during iteration with only an additive constant to alter the shape, which is why it has the more elaborate chaotic behavior vs. the IFS-like structure here.
Each fragment of a pattern roughly resembles a Cantor set, with fragments overlapping heavily except at the outer fringes. Each fragment is twisted according to the nature of complex multiplication, and the combination creates the similarity you noticed.
The latter two points are fairly intuitive given the nature of complex numbers and mostly apparent from inspection of the plots, but don't really do much to explain why, which I imagine is why Baez didn't remark on it.
Also unexplained are the gaps, such as those around the roots of unity. I don't recall the exact placements, but I recall it also being obvious why the locations of the gaps were relevant, but not why root density drops off so sharply around them. I could probably make some guesses, but only with copious handwaving involved.
On the other hand, except for the gaps, I believe the thick ring around the unit circle simply consists of more variations on the same patterns, denser and heavily overlapped, until detail is no longer visible.
Regarding variations on plots like this, the same approximate pattern appears regardless of polynomial degree, and for any set of coefficients of the form $\{-N, -(N - 1) ... -1, 1 ... N - 1, N\}$. The density of roots around the unit circle, the prominence of the gaps around certain points, and the character of the fringe inside the gaps all vary with degree; this is hard to see in Derbyshire's plot rather than one limited to polynomials of a single degree.
The shape of the overall plot changes if the coefficients are not chosen as above, being either of different magnitudes or chosen from more than one set. Note the plots on this page, particularly those near the bottom. The coloring indicates sensitivity of the points to changes in coefficients, which indicates which parts of the plot may distort asymmetrically.
My observations, in general, were:
Large differences in magnitude between coefficients makes the patterns tighter and denser, without changing the symmetry if each individual coefficient is chosen from a set like the original plot uses.
Large differences in magnitude between the elements in each set makes the plot as a whole more asymmetric, while making the density variations more prominent and removing the fringe patterns around the edge.
Doing both in moderation creates... odd effects:
Best Answer
It slightly depends on your definition of fractal, but typically no it is not true that IFS always generate fractals. Let $X = [0,1]$ and consider $f_1$ and $f_2$ defined on $X$ by $f_1(x) = \frac{x}{2}$ and $f_2(x) = \frac{1+x}{2}$. These are both contractions with a contractive factor of $\frac{1}{2}$. It's easy to check that $$ [0,1] = f_1([0,1]) \cup f_2([0,1]),$$ so the invariant set is $X$ itself.
There's also the dumb example where the IFS consists of a single contraction, then the attractor is just a point.