  Non-Hermitian systems exhibit interesting band structures, where novel
topological phenomena arise from the existence of exceptional points at which
eigenvalues and eigenvectors coalesce. One important open question is how this
would manifest at non-integer dimension. Here, we report on the appearance of
fractal eigenvalue degeneracies and Fermi surfaces in Hermitian and
non-Hermitian topological band structures. This might have profound
implications on the physics of black holes and Fermi surface instability driven
phenomena, such as superconductivity and charge density waves.
