Chaos at the rim of black hole and fuzzball shadows

We study the scattering of massless probes in the vicinity of the photon-sphere of asymptotically AdS black holes and horizon-free microstate geometries (fuzzballs). We find that these exhibit a chaotic behaviour characterised by exponentially large deviations of nearby trajectories. We compute the Lyapunov exponent lambda governing the exponential growth in d dimensions and show that it is bounded from above by lambda b=d-3/2bmin where b(min) is the minimal impact parameter under which a massless particle is swallowed by the black hole or gets trapped in the fuzzball for a very long time. Moreover we observe that lambda is typically below the advocated bound on chaos lambda(H) = 2 pi kappa T-B / , that in turn characterises the radial fall into the horizon, but the bound is violated in a narrow window near extremality, where the photon-sphere coalesces with the horizon. Finally, we find that fuzzballs are characterised by Lyapunov exponents smaller than those of the corresponding BH's suggesting the possibility of discriminating the existence of micro-structures at horizon scales via the detection of ring-down modes with time scales lambda(-1) longer than those expected for a BH of the given mass and spin.