  The relationship of Choptuik scaling to the scale invariance of Einstein's
equation is explored. Ordinary dynamical systems often have limit cycles:
periodic orbits that are the asymptotic limit of generic solutions. We show how
to separate Einstein's equation into the dynamics of the overall scale and the
dynamics of the ``scale invariant'' part of the metric. Periodicity of the
scale invariant part implies periodic self-similarity of the spacetime. We also
analyze a toy model that exhibits many of the features of Choptuik scaling.
