For each N, is there an N×N invertible matrix T over ℤ/2ℤ which does not have a stable subspace of "even weight" -- i.e. such that there does not exist a set of vectors over ℤ/2ℤ which all have an even number of 1s, and which span a space which is preserved under the action of T?
Equivalently (I think): is there an N×N unimodular matrix T over the integers, for each N, which "eventually" (by applying it enough times, possibly zero) maps each integer vector with at least one odd coefficient to a vector with odd 1-norm?
I'm most interested in the case where N is a power of 2, but remarks on the general case would be interesting.