A growth gap for diffeomorphisms of the interval
  Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of its n-th iteration. We get a function of n called the growth sequence. Its asymptotic behaviour is an interesting invariant which naturally appears both in geometry of the diffeomorphisms groups and in smooth dynamics. Our main result is the following Gap Theorem: the growth rate of this sequence is either exponential, or at most quadratic with n. Further, we construct diffeomorphisms with quite an irregular behaviour of the growth sequence. This construction easily extends to arbitrary manifolds. 
Leonid Polterovich and Mikhail Sodin