  We study the asymptotic expansion of smooth one-dimensional maps. We give an
example of an interval map for which the optimal shrinking of components
exponential rate is not attained for any neighborhood of a certain fixed point
in the boundary of a periodic Fatou component. We prove a general result
asserting that, when this happens the components do shrink exponentially,
although the rate is not the optimal one. Finally, we give an example of a
polynomial with real coefficients, such that all its critical points in the
complex plane are real, and such that its asymptotic expansion as a complex map
is strictly smaller than its asymptotic expansion as a real map.
