The Amalgamation Property for automorphisms of ordered abelian groups

We prove that the category of ordered abelian groups equipped with an automorphism has the Amalgamation Property, deduce that their inductive theory is NIP in the sense of positive logic, and initiate a development of the latter framework.
As byproducts of the proof, we obtain a generalised version of the Hahn Embedding Theorem which allows to lift each automorphism of an ordered abelian group to one of an ordered real vector space, and we show that, on existentially closed structures, linear combinations of iterates of the automorphism have the Intermediate Value Property.