  We study the problem of conjugating a diffeomorphism of the interval to
(positive) powers of itself. Although this is always possible for
homeomorphisms, the smooth setting is rather interesting. Besides the obvious
obstruction given by hyperbolic fixed points, several other aspects need to be
considered. As concrete results we show that, in class $C^1$, if we restrict to
the (closed) subset of diffeomorphisms having only parabolic fixed points, then
the set of diffeomorphisms that are conjugate to their powers is dense, but its
complement is generic. In higher regularity, however, the complementary set
contains an open and dense set. The text is complemented with several remarks
and results concerning distortion elements of the group of diffeomorphisms of
the interval in several regularities.
