Iterated function systems that generates Cantor Dust set

Question

I am wondering about the different iterated function systems which generates the Cantor Dust set.

I know that by definition, we call $X:\phi_1, \phi_2,...,\phi_m$ an iterated function system if $X$ is a closed subset of $\mathbb{R}^n, m\in\mathbb{N}$ and $\phi_1, \phi_2,...,\phi_m: X\to X $ are all contractions.

And a contraction is as follows. For metric spaces $(X,d)$ and $(X',d')$, a mapping $\phi:X\to \ X'$ is called a contraction if there exist another number $0\leq c < 1$, such that $d'(\phi(x),\phi(y))\leq cd(x,y)$ $ \forall x,y\in\ X $.

How can I use the definitions to find different iteration functions that generates Cantor Dust set?

Answer 1

The standard "middle thirds" Cantor set is the first example that everyone sees, and it is generated by the iterated function system with $X = [0,1]$, $\phi_1(x) = \frac{x}{3}$, and $\phi_2(x)=\frac{x+2}{3}$.

Besides the conditions on $X,\phi_1,...,\phi_m$ that are listed in your question, there are a few others that you should assume:

• $X$ is compact;
• Each of the functions $\phi_i$ is injective on $X$;
• Any two of the subsets $\phi_1(X),...,\phi_m(X)$ are disjoint from each other.

With these properties, you are guaranteed that this iterated function system generates a Cantor set, that set being defined as $$X_0 \cap X_1 \cap X_2 \cap \cdots $$ where the $X_j$'s are defined inductively by $X_0=X$ and $X_j = \cup_{i=1}^m \phi_i(X_{j-1})$.

It is utterly impossible to "know" ALL of the different iterated function systems that satisfy these properties.

But on the other hand, it is as easy as pie to produce zillions of different examples in $\mathbb R^n$ that satisfy these conditions. Here are some such examples.

Let $X \subset \mathbb R^n$ be any closed ball. Let $B_1,...,B_m \subset X$ be any collection of proper sub-balls (hence of strictly smaller radius than $X$) such that any two of them are disjoint. Let $\phi_i : X \to B_i$ be any collection of similarity maps, one for each $i=1,...,m$; this means that $\phi_i$ takes the center of $X$ to the center of $B_i$, and $\phi_i$ contracts distance by a factor equal to the ratio between the radius of $B_i$ and the radius of $X$. Then the iterated function system $X : \phi_1,...,\phi_m$ satisfies the conditions of your question and the additional conditions I listed, and it generates a Cantor set.