# Integer IFS

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Let $A$ - invertible integer matrix $n\times n$, $S=\{S_1,S_2,..S_k\}$ - finite group of integer matrices $n\times n$, such that $AS=SA$ (symmetry group of $A$).

Let $P$ - matrix $m\times n$ (projector from $R^n$ to $R^m$), $B$ - matrix $m\times m$, $G=\{G_1,G_2,..G_k\}$ - matrices $m\times m$ such that:

1. $\rho(B^{-1})\lt 1$ (see Spectral Radius)
2. $PA=BP$
3. $PS_i=G_iP$

Let $f_i(x)=G_{k_i}B^{-d_i}(x+P t_i)$, where $d_i\gt 0$ and $t_i$ is an integer vector from $R^n$.

We call any such IFS $\{f_1,f_2,.. f_n\}$ an Integer IFS.

It easily to extend this definition to directed graph IFS and rational numbers.

Many famous IFS can be represented as Integer: