  We study the scattering resonances of one-dimensional deterministic aperiodic
chains of electric dipoles using the vectorial Green's matrix method, which
accounts for both short- and long-range electromagnetic interactions in open
scattering systems. We discover the existence of edge-localized scattering
states within fractal energy gaps with characteristic topological band
structures. Notably, we report and characterize edge-localized modes in the
classical wave analogues of the Su-Schrieffer-Heeger (SHH) dimer model,
quasiperiodic Harper and Fibonacci crystals, as well as in more complex
Thue-Morse aperiodic systems. Our study demonstrates that topological
edge-modes with characteristic power-law envelope appear in open aperiodic
systems and coexist with traditional exponentially localized ones. Our results
extend the concept of topological states to the scattering resonances of
complex open systems with aperiodic order, thus providing an important step
towards the predictive design of topological optical metamaterials and devices
beyond tightbinding models.
