  We investigate the problem of succinctly representing an arbitrary
permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for
any i and any (positive or negative) integer power k. A representation taking
(1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in
constant time, for any positive constant \epsilon <= 1. A representation taking
the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers
in O(lg n / lg lg n) time.
  We then consider the more general problem of succinctly representing an
arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed
quickly for any i and any integer power k. We give a representation that takes
(1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and
computes arbitrary positive powers in constant time. It can also be used to
compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time.
  We place emphasis on the redundancy, or the space beyond the
information-theoretic lower bound that the data structure uses in order to
support operations efficiently. A number of lower bounds have recently been
shown on the redundancy of data structures. These lower bounds confirm the
space-time optimality of some of our solutions. Furthermore, the redundancy of
one of our structures "surpasses" a recent lower bound by Golynski [Golynski,
SODA 2009], thus demonstrating the limitations of this lower bound.
