Recall definition of [[Iterated function systems]]. # Statements Let $S$ be a [[complete metric spaces|complete metric space]]; $(r_{1},r_{2},\dots,r_{n})$ a contracting ratio list (meaning $r_{i}<1$); and $(f_{1},f_{2},\dots,f_{n})$ an iterated function system consisting of Lipschitz continuous functions $f_{i}$ with Lipschitz constant $r_{i}$ respectively. Then there exists a unique non-empty *compact* invariant set $A$ for the iterated function system, i.e. $A= \bigcup_{i=1}^{n} f_{i}(A)$ (Edgar, Theorem 4.1.3 and Exercise 4.1.5). May also be credited to Moran-Hutchison. # Applications # Proofs ###### Sketch of proof Let $\mathbb{H}(S)$ denote the set of non-empty compact subset and $D:\mathbb{H}(S) \times \mathbb{H}(S)\to [0,\infty)$ denote [[Hausdorff distance|Hausdorff distance]] which is defined by $D(A,B) = \inf \{ r>0: N_{r}(A) \subseteq B \text{ and } N_{r}(B) \subseteq A \}.$ Define $F$ on $\mathbb{H}(S)$ by $F(A) = \bigcup_{i=1}^{n} f_{i}(A)$. By [[Space of non-empty compact subsets of complete metric space is complete wrt Hausdorff distance]], $\mathbb{H}(S)$ is complete since $S$ is complete so we want to show that $F$ is a contraction so that the stated result then follows from the [[contraction mapping theorem]]. In particular, we will show that $D(F(A), F(B))\leq r D(A,B)$ where $r=\max_{1\leq i\leq n} r_{i}$. Since limit preserve weak inequalities, it suffices to show that for all $q>D(A,B)$, we have that $D(F(A), F(B))\leq rq$. By unpacking the definition of $D(A,B)$, $q>D(A,B)$ yields a proof that $F(A)$ is in the $rq$-neighbourhood of $F(B)$ and vice-versa which completes the proof.