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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.

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  • $\begingroup$ Are the $W_i$ fixed beforehand? Or else, how are they chosen? Also, maybe say explicitly that these are matrices with complex entries. The assumption that $A,B,C$ they are not unitary is a bit weird since it breaks the conjugacy-invariance of the problem: do you simply mean "not necessarily unitary"?. $\endgroup$
    – YCor
    Commented Mar 28, 2022 at 7:33
  • $\begingroup$ The words are fixed beforehand, and I would like to find out what matrices $A,B,C$ lead us to bounded norms of products $\endgroup$
    – Станислав Крымский
    Commented Apr 1, 2022 at 15:59
  • $\begingroup$ If the three matrices are unitary, they inevitably lead us to the bounded norm $\endgroup$
    – Станислав Крымский
    Commented Apr 1, 2022 at 16:00

1 Answer 1

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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.

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  • $\begingroup$ I am afraid the setup of random products is quite different from that of OP's question $\endgroup$
    – R W
    Commented Mar 27, 2022 at 22:44
  • $\begingroup$ @R W It depends if the W_i are fixed once for all, or one picks them afresh in each step. The first one is a substitution dynamical system, while the second, more general, option includes random products. $\endgroup$
    – Yuval Peres
    Commented Mar 28, 2022 at 3:01
  • $\begingroup$ Even in the model you describe the arising random products are not products with stationary (let alone independent) increments. At time $n$ you will have products of the original matrices of length $2^n$. $\endgroup$
    – R W
    Commented Mar 28, 2022 at 10:26

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