[darkmighty]

If you look at the image of geodesics he shows an example of an almost 360 degree trajectory. Simple continuity arguments allow you to conclude you can have an arbitrary number of rotations, I believe.

I would conjecture the contraction is roughly exponential, there's no obvious contradiction with that but it's way over by knowledge to check it ...

[wyager]

By the time a photon has done a complete rotation, it necessarily has to be on a path that takes it into the black hole (instead of outside of it). So yes, you can have an arbitrary number of rotations, but anything that makes a complete rotation or more will never leave the black hole.

[darkmighty]

It turns out that a photon can describe any real rotation angle around a black hole (I've asked a physicist). The continuity argument I gave is valid: take the critical energy E wherein the photon is attached to the photosphere, then with an adequate E+eps, eps > 0, the photon will describe an according monotonically decreasing angle.

[darkmighty]

Hmm I'm not so sure anymore. I'll have to ask a physicist I guess.