  We investigate the existence of interface states induced by broken inversion
symmetries in two-dimensional quasicrystal lattices. We introduce a 10-fold
rotationally symmetric quasicrystal lattice whose inversion symmetry is broken
through a mass dimerization that produces two 5-fold symmetric sub-lattices. By
considering resonator scatterers attached to an elastic plate, we illustrate
the emergence of bands of interface states that accompany a band inversion of
the quasicrystal spectrum as a function of the dimerization parameter. These
bands are filled by modes which are localized along domain-wall interfaces
separating regions of opposite inversion symmetry. These features draw
parallels to the dynamic behavior of topological interface states in the
context of the valley Hall effect, which has been so far limited to periodic
lattices. We numerically and experimentally demonstrate wave-guiding in a
quasicrystal lattice featuring a zig-zag interface with sharp turns of 36
degrees, which goes beyond the limitation of 60 degrees associated with 6-fold
symmetric (i.e., honeycomb) periodic lattices. Our results provide new
opportunities for symmetry-based quasicrystalline topological waveguides that
do not require time-reversal symmetry breaking, and that allow for higher
freedom in the design of their waveguiding trajectories by leveraging
higher-order rotational symmetries.
