Growth Gap vs. smoothness for diffeomorphisms of the interval

TL;DR

This paper investigates the asymptotic behavior of the growth sequence of diffeomorphisms of the interval, providing sharp estimates based on the modulus of continuity of the derivative, thus advancing understanding in smooth dynamics and geometric group theory.

Contribution

It extends previous results by establishing sharp estimates for the growth sequence using the modulus of continuity, connecting smooth dynamics with geometric properties of diffeomorphism groups.

Findings

Sharp estimates for growth sequences derived

Extensions of Polterovich, Sodin, and Borichev's results

Enhanced understanding of diffeomorphism group geometry

Abstract

Given a diffeomorphism of the interval, consider the uniform norm of the derivative of its n-th iteration. We get a sequence of real numbers called the growth sequence. Its asymptotic behavior is an invariant which naturally appears both in smooth dynamics and in geometry of the diffeomorphisms groups. We find sharp estimates for the growth sequence of a given diffeomorphism in terms of the modulus of continuity of its derivative. These estimates extend previous results of Polterovich and Sodin, and Borichev.