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]