  We show that scalar, electromagnetic, and gravitational perturbations of
extremal Kerr black holes are asymptotically self-similar under the
near-horizon, late-time scaling symmetry of the background metric. This
accounts for the Aretakis instability (growth of transverse derivatives) as a
critical phenomenon associated with the emergent symmetry. We compute the
critical exponent of each mode, which is equivalent to its decay rate. It
follows from symmetry arguments that, despite the growth of transverse
derivatives, all generally covariant scalar quantities decay to zero.
