Skip to main content
MathOverflow

Return to Answer

added 342 characters in body
Source Link
Yuval Peres
Yuval Peres
  • 14.2k
  • 1
  • 28
  • 49

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.

Suppose you form the words $W_i$ 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.

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.

Source Link
Yuval Peres
Yuval Peres
  • 14.2k
  • 1
  • 28
  • 49

Suppose you form the words $W_i$ 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.