Background
Let $\psi_1, \psi_2, ... \psi_n :\mathbb{R}^n\to\mathbb{R}^n$ be similarity mappings defining an iterated function system $\psi$. To each $\psi_i$ we assign the similarity coefficient $r_i \in \mathbb{R}$. By assumption, the mappings are contractive so each $r_i<1$.
The mappings are contractive, so let $V$ be the unique fixed point of the iterated function system $\psi$.
Furthermore, suppose $\psi$ satisfies the open set condition. Then the following equation holds $$\sum_{i=1}^n r_i^d=1 $$ where $d$ is the Hausdorff dimension of $V$. This identity is sometimes referred to as the Moran equation (e.g. this answer).
Question
I am interested in a converse statement of the following form:
- Let $d,r_1,...r_n\in\mathbb{R}$ satisfy the Moran equation. When does there exist an iterated function system to which this equation corresponds?
If $d=1$, then there always exists such a system.
If $d=2$, then consider a solution to the Moran equation $r_1^2+r_2^2=1$. Then there is an iterated function system corresponding to the right-angled triangle with sides $1,r_1,r_2$, which has $r_1,r_2$ as its coefficients of similarity.
This leads me to ask the following question:
If $d\in\mathbb{N}$ and $r_1,r_2$ are positive real solutions to $r_1^d+r_2^d=1$, does there exist an iterated function system whose similarity coefficients are $r_1, r_2$ and whose attractor has Hausdorff dimension $d>0$?
(I have now asked a simplified version of this question, with illustrations of the geometry, here.)
