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The interplay between incommensurability and strong correlations is a challenging open issue. It is explored here via numerical renormalization-group (NRG) study of models of a magnetic impurity in a one-dimensional quasicrystal. The principal goal is to elucidate the physics at the localization transition of the Aubry-André Hamiltonian, where a fractal spectrum and multifractal wave functions lead to a critical Aubry-André Anderson (AAA) impurity model with an energy-dependent multifractal hybridization function. This goal is reached in three stages of increasing complexity: (1) Anderson impurity models with uniform fractal hybridization functions are solved to arbitrarily low temperatures . Below a Kondo temperature, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties are oscillatory in about negative average values determined by the spectrum's fractal dimension , with …