We investigate the qualitative features of binary black hole shadows using the model of two
extremally charged black holes in static equilibrium (a Majumdar–Papapetrou solution). Our
perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they
admit more than one fundamental null orbit, and thus an uncountably infinite set of perpetual null
orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we
develop an appropriate symbolic dynamics to describe planar null geodesics on the double black
hole spacetime. We show that a one-dimensional (1D) black hole shadow may constructed through
an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar.
Next, we study non-planar rays, to understand how angular momentum affects the existence and
properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three
types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to
regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly
chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist
around two black holes of equal mass M separated by a1 < a < √
2a1, where a1 = 4M/√
27. To
show how this possibility arises, we define a certain potential function and classify its stationary
points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property.
Finally, we consider the possibility of following null geodesics through event horizons, and chaos in
the maximally extended spacetime
