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

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 the absence of such fixed points, the maximal growth of the derivatives of the iterates is subquadratic.