To understand the effect of forbidden combinations, we introduce addresses of regions of the unit square S.
If some combinations of transformations are not allowed to occur, then no points land in regions with addresses containing some sequences. First, here are some simple examples.
T4 never immediately follows T1.
These examples illustrate the general situation: if some
combination of
The transformation combinations that never occur are called forbidden combinations. For example, if T4 never immediately follows T1, we say 41 is a forbidden pair. We have just seen that any triple containing a forbidden pair is a forbidden triple. These triples are "automatically" forbidden, but there are other ways for a triple to be forbidden. Specifically, there are Driven IFS having forbidden triples containing no forbidden pairs, forbidden quadruples containing no forbidden triples, and so on. One measure of the complexity of a fractal is the size of the smallest collection of forbidden combinations necessary to produce the fractal.
For those Driven IFS determined completely by forbidden pairs, a compact representation of the IFS can be given by a graph showing the allowed pairs. The graph has four vertices, one for each Ti, and an edge from vertex i to vertex j if Ti can be be followed immediately by Tj. For example, the Driven IFS with the single forbidden pair 41 has this graph:
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Because T1 cannot be immediately followed by T4, the graph has no arrow from vertex 1 to vertex 4. All other combinations are allowed, so all other pairs of edges are connected by arrows.
Different combinations of forbidden pairs can generate some very interesting pictures. Here are some examples.
Before departing from Driven IFS determined by forbidden pairs, we consider a natural question: when can the picture generated by a Driven IFS determined by forbidden pairs also be generated by an IFS with no forbidden combinations and all transformations similarities, but perhaps with more than four transformations? The answer can be read from the graph.
Finally, we make a brief comment about higher-order exclusions.
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