  It is expected that black holes are formed dynamically under the
gravitational collapses and approach to the stationary states. In this paper,
we show that the asymptotic Killing vector at late time should exist on the
horizon and then that it can be extended to the out of black holes under the
assumption of the analyticity of spacetimes. This fact implies that if there is
another asymptotic Killing vector which becomes stationary Killing at far
region and spacelike in "ergoregion", the rotating black holes may have the
asymptotically axisymmetric Killing vector at late time. Thus, we may expect
that the asymptotic rigidity of the black holes holds.
