  We investigate the nonequilibrium response of quasiperiodic systems to
boundary driving. In particular we focus on the Aubry-Andr\'e-Harper model at
its metal-insulator transition and the diagonal Fibonacci model. We find that
opening the system at the boundaries provides a transparent and experimentally
viable technique to probe its underlying fractality, which is reflected in the
fractal spatial dependence of simple observables (such as magnetization) in the
nonequilibrium steady state. We also find that the dynamics in the
nonequilibrium steady state depends on the length of the chain chosen: generic
length chains harbour qualitatively slower transport (different scaling
exponent) than Fibonacci length chains, which is in turn slower than in the
closed system. We conjecture that such fractal nonequilibrium steady states
should arise in generic driven critical systems that have fractal properties.
