Driven IFS with Forbidden Combinations

To understand the effect of forbidden combinations, we use the addresses of regions of the unit square S.

Examples can be explored this software.

We begin with three simple examples. Forbid a transformation: T4 never occurs. Forbid a pair: T4 never immediately follows T1. Here is another example of a forbidden pair. These examples illustrate the general situation: if some combination of Tin...Ti1 of transformations never occurs, then all subsquares with addresses containing the sequence in...i1 must be empty. 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. Graphical representation of forbidden pairs Software representation of forbidden pairs Different combinations of forbidden pairs can generate some very interesting pictures. Here are some examples. 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. Any triple containing a forbidden pair is a forbidden triple, but there also 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. Here are some examples with forbidden triples.

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