Does the set of matrices with bounded recursive products form a fractal? We are given three matrices $A,B,C$ which have determinant 1, are not unitary and that their size is the same. Consider the following process.
On each step we take three words $W_1,W_2,W_3$ consisting of letters $A,B,C$ and simultaneously replace the matrix $A$ with the product of matrices in the word $W_1$, the matrix $B$ with the product of $W_2$, the matric $C$ with $W_3$.
Consider the set of matrices $A,B,C$ such that all the matrices obtaied in the process have a bounded norm. What can be said of the set's properties? I think that it should have measure zero and behave like a fractal.
 A: There are two interpretations of the OP's question. If the W_i are fixed once for all, or one is allowed to pick them afresh in each step. In The first interpretation we obtain a substitution dynamical system (see e.g. https://link.springer.com/book/10.1007/978-3-642-11212-6). The second  option includes random products. Suppose you form the words $W_i$ (of length 2, say) at random, by taking each letter to be $A,B$ or $C$ with probability $1/3$, independently of previous choices. Then we are in the realm of random matrix products, where the Theorem of Furstenberg [1, theorem 2.1] applies.
It implies that the norm of this random product will grow exponentially unless either $A,B,C$ generate a compact group, or there is a finite union of hyperplanes that is preserved by $A,B,C$, both of these conditions define sets of measure zero.
There is a friendly exposition in the book [2], see theorem 6.3 page 66 there.
[1] Furstenberg, Harry. "Noncommuting random products." Transactions of the American Mathematical Society 108, no. 3 (1963): 377-428.
[2] Bougerol, Philippe and Jean Lacroix. Products of random matrices with applications to Schrödinger operators. Vol. 8. Springer Science & Business Media, 2012.
