A growth gap for diffeomorphisms of the interval

Details A growth gap for diffeomorphisms of the interval A growth gap for diffeomorphisms of the interval

2002-03-07

arxiv.org/abs/math/0203066v2

Journal d'Analyse Math. 92 (2004) pp. 191-209

Mathematics

Dynamical Systems

LATeX, 24 pp, incorporates DS/0110306, the latest version

Scientific paper

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.

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