  Fractal lattices, with their self-similar and intricate structures, offer
potential platforms for engineering physical properties on the nanoscale and
also for realizing and manipulating high order topological insulator states in
novel ways. Here we present a theoretical discussion on localized corner and
edge states, as well as electronic properties emerging from topological phases
in Sierpinski Carpet within a $\pi$-flux regime. A topological hopping
parameter phase diagram is constructed from which different spatial localized
states are identified following signatures of distinct fractal generations. The
specific geometry and scaling properties of the fractal systems can guide the
supported topological states types and their associated functionalities. A
conductive device is proposed by coupling identical Sierpinski Carpet units
providing transport response through projected edge states that carrier the
details of the Sierpinski Carpet topology.
