The rate of growth of a matrix semigroup is often quite delicate. See http://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-01145-X/S0025-571... for a fun example. Victor Sent from my iPhone
On Mar 21, 2014, at 14:35, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Sorry everybody --- I fouled up by oversimplifying the problem.
So let's make it a semigroup generated by a given finite set of integer matrices (it seems experimentally that my constraints can be jettisoned), and I'm trying to show that every product of s generators has all eigenvalues bounded in modulus by c^s (where c is also given).
A toy example, which I can actually decide by elementary means, is generated by 10 2x2 matrices shown below, where c = (1 + rt5)/2 is the golden section.
Notice that the product
[1 0] [1 1] = [1 1] [1 1] [0 1] = [1 2]
has eigenvalue c^2 , despite both generators having unit eigenvalues.
WFL
___________________________
matOOOO; [1 1] [1 0] matOOOI; [1 0] [1 0] matOOII; [1 0] [1 1] matOIOO; [1 1] [0 0] matOIOI; [1 0] [0 1] matOIIO; [0 1] [1 0] matOIII; [0 0] [1 1] matIIOO; [1 1] [0 1] matIIOI; [0 1] [0 1] matIIII; [0 1] [1 1]
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