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We calculate the electronic structure in quasiperiodic double-moir\'e systemsof graphene sandwiched by hexagonal boron nitride, and identify the topologicalinvariants of energy gaps. We find that the electronic spectrum contains anumber of minigaps, and they exhibit a recursive fractal structure similar tothe Hofstadter butterfly when plotted against the twist angle. Each of theenergy gaps can be characterized by a set of integers, which are associatedwith an area in the momentum space. The corresponding area is geometricallyinterpreted as a quasi Brillouin zone, which is a polygon enclosed by multipleBragg planes of the composite periods and can be uniquely specified by theplain wave projection in the weak potential limit.

Fractal energy gaps and topological invariants in hBN/Graphene/hBN double moir\'{e} systems