  We investigate the dynamics of the well-known RSK bijection on permutations
when iterated on various reading words of the recording tableau. In the setting
of the ordinary (row) reading word, we show that there is exactly one fixed
point per partition shape, and that it is always reached within two steps from
any starting permutation.
  We also consider the modified dynamical systems formed by iterating RSK on
the column reading word and the reversed reading word of the recording tableau.
We show that the column reading word gives similar dynamics to the row reading
word. On the other hand, for the reversed reading word, we always reach either
a 2-cycle or fixed point after two steps. In fact, we reach a fixed point if
and only if the shape of the initial tableau is self-conjugate.
