Are the invariant sets of all iterated function systems necessarily fractal?

Question

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.

Answer 1

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.