Focusing and the holographic hypothesis

The "screen mapping" introduced by Susskind to implement 't Hooft's holographic hypothesis is studied. For a single screen time, there are an infinite number of images of a black-hole event horizon, almost all of which have a smaller area on the screen than the horizon area. This is consistent with the focusing equation because of the existence of focal points. However, the boundary of the past (or future) of the screen obeys the area theorem, and so always gives an expanding map to the screen, as required by the holographic hypothesis. These considerations are illustrated with several axisymmetric static black-hole spacetimes.