Exact quadratic growth for the derivatives of iterates of interval diffeomorphisms with only parabolic fixed points

TL;DR

This paper establishes that for certain interval diffeomorphisms with parabolic fixed points, the derivatives of their iterates grow exactly quadratically under specific conditions, and subquadratically otherwise.

Contribution

It provides a precise characterization of the growth rate of derivatives for $C^2$ interval diffeomorphisms with parabolic fixed points, distinguishing between quadratic and subquadratic growth.

Findings

Maximal derivative growth is exactly quadratic with non-quadratic tangency.

Without such fixed points, growth is subquadratic.

Growth behavior depends on the nature of fixed points and tangencies.

Abstract

We consider $C^2$ diffeomorphisms of a closed interval with only parabolic fixed points. We show that the maximal growth of the derivatives of the iterates of such a diffeomorphism is exactly quadratic provided it has a non-quadratical tangency to the identity at a fixed point that is topologically repelling on one side. Moreover, in absence of such fixed points, the maximal growth of the derivatives of the iterates is subquadratic.