Integer IFS

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Let [math]A[/math] - invertible integer matrix [math]n\times n[/math].

[math]S=\{S_1,S_2,..S_k\}[/math] - group of integer matrices [math]n\times n[/math], such that [math]AS=SA[/math] (symmetry group of [math]A[/math]).

Let [math]P[/math] - matrix [math]m\times n[/math] (projector from [math]R^n[/math] to [math]R^m[/math]),

[math]B[/math] - matrix [math]m\times m[/math], [math]R=\{R_1,R_2,..R_k\} [/math] - matrices [math]m\times m[/math]

such that:

  1. [math]|B^{-1}|\lt 1[/math]
  2. [math]P\times A=B\times P[/math]
  3. [math]P\times S_i=R_i\times P[/math]
  4. [math]B[/math] and [math]\{R_i\}[/math] is in a real Jordan normal form


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

We call any such IFS [math]\{f_1,f_2,.. f_n\}[/math] an Integer IFS. (see Iterated function system)

It easily to extend this definition to directed graph IFS.

Many famous IFS can be represented as Integer:

Sierpinski triangle

Rauzy fractal

Dragon curve

Golden Bee

Pentadentrite

McWorters Pentigree

References

Self-Similar Sets 5. Integer Matrices and Fractal Tilings of Rn