2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Do you require a separation condition? If not, I guess it's easy to come up with an example. $\endgroup$ Feb 19, 2016 at 18:22
  • $\begingroup$ Yes, it is easy. I will edit my question at night. $\endgroup$
    – YTS
    Feb 19, 2016 at 20:12

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.