  We consider the one-dimensional $XX$-model in a quasi-periodic
transverse-field described by the Harper potential, which is equivalent to a
tight-binding model of spinless fermions with a quasi-periodic chemical
potential. For weak transverse field (chemical potential), $h<h_c$, the
excitations (fermions) are delocalized, but become localized for $h>h_c$. We
study the non-equilibrium relaxation of the system by applying two protocols: a
sudden change of $h$ (quench dynamics) and a slow change of $h$ in time
(adiabatic dynamics). For a quench into the delocalized (localized) phase, the
entanglement entropy grows linearly (saturates) and the order parameter
decreases exponentially (has a finite limiting value). For a critical quench
the entropy increases algebraically with time, whereas the order parameter
decreases with a stretched-exponential. The density of defects after an
adiabatic field change through the critical point is shown to scale with a
power of the rate of field change and a scaling relation for the exponent is
derived.
