Iterated Function Systems as Automata Networks

When we want to realize the dynamics of a digitized IFS

F =((X,d), f₁, f₂, ..., f_k)

with the help of an automata network $\sigma$, we have the following picture.

As C we take a regular lattice of some dimensionality m. In a practically interesting case, m = 2 and $C = \{(i,j) \,\vert\, i = 1, \dots, p, \, j = 1, \dots, q\}$. The alphabet of states $A = \{0, 1\}$. The cell neighborhoods are defined by the inverses of the IFS's functions:

$$\forall c \in C: \;\; N(c) = \bigcup_{i=1}^k f_i^{-1}(c).$$ (15)

The LDRs $\delta_c$ are the same for all cells c:

$$\forall c \in C: \;\; \delta_c(S_{N(c)}) = \left\{ \begin{ar... ...\prime) = 1 \\ 0 & \mbox{\rm otherwise} \end{array} \right.$$ (16)

To build a bridge to our neural network algorithm (Sec. 4.3), we take as C our unit square S^u of pixels and obtain the neighborhoods N(c) by analogy with computing the synaptic weights.

The whole process of going from an initial IFS

$$F =((X,d), f_1, f_2, ..., f_k) \qquad \& \qquad F=\bigcup_{i=1}^k f_i$$

to the corresponding automata network $\sigma$ looks like the following.

Fist of all we go to the dynamical system

$$(({\cal H}(X),h),F),$$

then, after digitization and scaling, to the dynamical system

$$(({\cal H}(S^u),h),F),$$

where S^u is the unit square.

Then we use our Automata Network Algorithm to get a special intermediate form of the IFS F which we call "geometrical form":

$$F_g=\{<s_0^i,s_t^i,s_r^i>\}_{i \in \{1,2, \dots, k\}}$$

where s₀ⁱ,s_tⁱ,s_rⁱ are the images of the corner points s₀,s_t,s_r under $f_i \in F$ (Sec. 4.3).

We continue to apply the Automata Network Algorithm and eventually construct the automata network

$$(S^u,\{0,1\},N,A_0,\Delta) \qquad \& \qquad N=\bigcup_{s \in S^u}\bigcup_{f \in F}f^{-1}(s).$$

We start the dynamical process, beginning from a compact set $A_0 \subseteq S^u$, and after a big enough number M of iterations get an approximation A_M of the IFS's attractor A_F:

$$\Delta^*: A_0 \longrightarrow A_M \approx A_F.$$