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 diffeomorphism 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 whose growth sequence has quite irregular behaviour. This construction easily extends to arbitrary manifolds.