3 rescaling is bogus.; added 46 characters in body

(Oops, the rescaling part is bogus in the below. So this only works for C with determinant 1.)

In the 2-by-2 case, the answer is no. (Something like this argument should go through in general).

After rescaling, we can assume the matrix has determinant 1. If C is elliptic (real trace between -2 and 2), then all powers are elliptic, so that's no good. If it's parabolic (trace equal to -2 or 2), then all powers all parabolic, again no good. If it's loxodromic, the traces of the powers have real part going to infinity with n, and so they can't be dense.

2 rescaled matrix.

In the 2-by-2 case, the answer is no. (Something like this argument should go through in general).

After rescaling, we can assume the matrix has determinant 1. If C is elliptic (real trace between -2 and 2), then all powers are elliptic, so that's no good. If it's parabolic (trace equal to -2 or 2), then all powers all parabolic, again no good. If it's loxodromic, the traces of the powers have real part going to infinity with n, and so they can't be dense.

1

In the 2-by-2 case, the answer is no. (Something like this argument should go through in general).

If C is elliptic (real trace between -2 and 2), then all powers are elliptic, so that's no good. If it's parabolic (trace equal to -2 or 2), then all powers all parabolic, again no good. If it's loxodromic, the traces of the powers have real part going to infinity with n, and so they can't be dense.