  The electromagnetic signature of a point explosion near a Kerr black hole
(BH) is evaluated. The first repetitions produced by gravitational lensing are
not periodic in time; periodicity emerges only as the result of multiple
circuits of the prograde and retrograde light rings and is accompanied by
exponential dimming. Gravitational focusing creates a sequence of concentrated
caustic features and biases the detection of a repeating source toward
alignment of the BH spin with the plane of the sky. We consider the
polarization pattern in the case of emission by the Lorentz upboosting and
reflection of a magnetic field near the explosion site. Then the polarized
fraction of the detected pulse approaches unity, and rays propagating near the
equatorial plane maintain a consistent polarization direction. Near a slowly
accreting supermassive BH (SMBH), additional repetitions are caused by
reflection off annular fragments of an orbiting disk that has passed through an
ionization instability. These results are applied to the repeating fast radio
burst (FRB) source 121102, giving a concrete and predictive example of how FRB
detectability may be biased by lensing. A gravitational lensing delay of 10-30
s, and reflection delay up to $\sim 10^4$ s, are found for emission near the
innermost stable circular orbit of a $3\times 10^5\,M_\odot$ SMBH; these
effects combine to produce interesting correlations between delay time and
burst fluence. A similar repetitive pulse envelope could be seen in the
gravitational wave signal produced by a collision between compact stars near a
SMBH.
