  We show that a $\mathrm{GL}(d,\mathbb{R})$ cocycle over a hyperbolic system
with constant periodic data has a dominated splitting whenever the periodic
data indicates it should. This implies global periodic data rigidity of generic
Anosov automorphisms of $\mathbb{T}^d$. Further, our approach also works when
the periodic data is narrow, that is, sufficiently close to constant. We can
show global periodic data rigidity for certain non-linear Anosov
diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with
simple spectrum.
