We study the three-dimensional structure formation when atoms are deposited onto a substrate with a decagonal quasicrystalline order. Molecular-dynamicscalculations show that the adsorbate layer consists of ordered nano-scale domains with orientations determined by the underlying substrate symmetry. Depending on the relative strength of the interactions of adsorbate atoms with each other and with the substrate atoms, different morphologies are observed ranging from layer-by-layer growth to cluster formation. We also find that the film thickness likewise affects the overall structure of the growing film: Depending on the relative strength of the interaction between adsorbate atoms, a structural transition of the configuration of the adsorbate layers closest to the substrate can occur as the number of deposited layers increases.