  We present Floquet fractal topological insulators: photonic topological
insulators in a fractal-dimensional lattice consisting of helical waveguides.
The helical modulation induces an artificial gauge field and leads to the
trivial-to-topological transition. The quasienergy spectrum shows the existence
of topological edge states corresponding to real-space Chern number 1. We study
the propagation of light along the outer edges of the fractal lattice and find
that wavepackets move along the edges without penetrating into the bulk or
backscattering even in the presence of disorder. In a similar vein, we find
that the inner edges of the fractal lattice also exhibit robust transport when
the fractal is of sufficiently high generation. Finally, we find topological
edge-states that span the circumference of a hybrid half-fractal half-honeycomb
lattice, passing from the edge of the honeycomb lattice to the edge of the
fractal structure virtually without scattering, despite the transition from
two-dimensions to a fractal dimension. Our system offers a realizable
experimental platform to study topological fractals and provides new directions
for exploring topological physics.
