Where do moving punctures go?

Currently the most popular method to evolve black-hole binaries is the ``moving puncture'' method. It has recently been shown that when puncture initial data for a Schwarzschild black hole are evolved using this method, the numerical slices quickly lose contact with the second asymptotically flat end, and end instead on a cylinder of finite Schwarzschild coordinate radius. These slices are stationary, meaning that their geometry does not evolve further. We will describe these results in the context of maximal slices, and present time-independent puncture-like data for the Schwarzschild spacetime.