  In this work, elastic fractal higher-order topological states are
investigated. Bott index is adopted to characterize the topological property of
elastic fractal structures. The topological corner and edge states of elastic
waves in fractal structures are realized theoretically and experimentally.
Different from traditional two-dimension (2D) high-order topological insulators
based on periodic structures, the high-order topological states based on
elastic fractal structures in this work support not only abundant topological
inner corner states and edge states, but also topological outer corner states.
The richness of corner states is much higher than that of topological
insulators based on periodic structures. The strong robustness of the
topological corner states in the fractal structure are verified by introducing
disorders and defects. The topological phenomenon of elastic fractal structures
revealed in this work enriches the topological physics of elastic systems and
breaks the limitation of that relies on periodic elastic structures. The
results have important application prospects in energy harvesting, information
transmissions, elastic energy acquisitions and high-sensitivity detections.
