# Critical set of a self similar structure

A self similar structure is a triple $L=(K, S, \{F_i\})$ where $K$ is a compact metric space, $S$ is a finite set and for every $i\in S$ the functions $F_i:K\rightarrow K$ are injective and continuous. In such a way that there exist a surjective function $\pi:\Sigma \rightarrow K$ and $\pi\circ \sigma_i=F_i\circ\pi$.

Here $\Sigma$ is the ultrametric space with letters $S$.

The set $C_L^K=\bigcup_{i\not= j}F_i(K)\cap F_j(K)$ is called the critical set of the self similar structure $L$. Each $F_i$ has exactly one fixed point. I want to know if there exists some self similar structure with its critical set containing some of the fixed points. The classical examples I have found do not; I want to know if this works in general.

EDIT: Indeed it is easy to find an example with these characteristics. We can choose $K=[-1,1]$ and the functions $F_1(x)=\frac{1}{2}x, F_2(x)=-\frac{1}{2}x, F_3(x)=\frac{1}{2}x+\frac{1}{2},F_4(x)=-\frac{1}{2}x-\frac{1}{2}$ then the point $0$ is a fixed point of $F_1$ and $F_2$.

Then I will impose an aditional condition. I want that the self similar structure be minimal.

All the notions I mentioned here are in Analysis on fractals (Kigami, 2001) in chapter 1.

• Do you require a separation condition? If not, I guess it's easy to come up with an example. – Mark McClure Feb 19 '16 at 18:22
• Yes, it is easy. I will edit my question at night. – Yotas Trejos Feb 19 '16 at 20:12