Filtered chaos

Suppose $S$ is a random sequence use to drive the chaos game. What is the effect of filtering $S$ to remove certain substrings?

The simplest case would be to choose a $1$-digit substring and remove it. Suppose the $1$-digit is string is $i_j$, then this filtering is indistinguishable from assigning $p_{i_j}$ $=$ 0 in generating the sequence. For example, if $\{w_1, \ldots, w_4\}$ were transformations corresponding to the full equilateral triangle, then filtering out all $4$s would generate Sierpinski's gasket.

I've implemented two methods of string filtering that you can experiment with at Fractal Sequence Applet The first method truncates the forbidden string: each occurrence of $i_1 \cdots i_k$ has $i_k$ removed. The second method extracts the forbidden string: each occurrence of $i_1 \cdots i_k$ has $i_1 \cdots i_k$ removed.

For concreteness, I focus on the Full Square IFS with substring ``12'' filtered out. This works well since we have some intuition about working with rectangular coordinates, so the addresses in the square have the form: lower-left-sub-square has address $1$, upper-left-sub-square has address $2$, upper-right-sub-square has address $3$, lower-right-sub-square has address $4$. This addressing pattern is iterated in the obvious way.