Iterated function system

Remarks

Hutchinson's theorem says that the attractor exists and is unique. The proof applies the Banach fixed point theorem to the map $K \mapsto \bigcup_i f_i(K)$ on the space of nonempty compact subsets of $X$ with the Hausdorff metric. This space is complete, and the map is a contraction on it with constant $\max_i r_i$. So for every nonempty compact $K_0$, the iterates $K_{m+1} = \bigcup_i f_i(K_m)$ converge to $A$.

Example: on $\R$, the maps $f_1(x) = x/2$ and $f_2(x) = x/2 + 1/2$ are contractions with constant $\frac{1}{2}$, and $f_1([0, 1]) \cup f_2([0, 1]) = [0, \frac{1}{2}] \cup [\frac{1}{2}, 1] = [0, 1]$. So the attractor is $[0, 1]$. With $f_1(x) = x/3$ and $f_2(x) = x/3 + 2/3$ the attractor is the Cantor set.

Non-example: the family $g_1(x) = x/3$, $g_2(x) = x + 1$ on $\R$ is not an iterated function system, since $g_2$ is not a contraction. It has no nonempty compact set $A$ with $A = g_1(A) \cup g_2(A)$: the maximum $a$ of such an $A$ would satisfy $a + 1 \in A$, which is impossible.

Suppose that the maps are similarities with ratios $r_i$ and that the pieces $f_i(A)$ do not overlap too much (the open set condition). Then the Hausdorff dimension of $A$ is the number $s$ with $\sum_i r_i^s = 1$.