  One of the most prominent characteristics of two-dimensional Quantum Hall
systems are chiral edge modes. Their existence is a consequence of the
bulk-boundary correspondence and their stability guarantees the quantization of
the transverse conductance. In this work, we study two microscopic models, the
Hofstadter lattice model and an extended version of Haldane's Chern insulator.
Both models host Quantum Hall phases in two dimensions. We transfer them to
lattice implementations of fractals with a dimension between one and two and
study the existence and robustness of their edge states. Our main observation
is that, contrary to their two-dimensional counterpart, there is no universal
behavior of the edge modes in fractals. Instead, their presence and stability
critically depends on details of the models and the lattice realization of the
fractal.
