NIP for the Asymptotic Couple of the Field of Logarithmic Transseries

The derivation on the differential-valued field ${\mathbb{T}}_{\log }$ of logarithmic transseries induces on its value group ${\mathrm{\Gamma }}_{\log }$ a certain map $\psi$. The structure $\mathrm{\Gamma }=({\mathrm{\Gamma }}_{\log },\psi )$ is a divisible asymptotic couple. In~\cite{gehret} we began a study of the first-order theory of $({\mathrm{\Gamma }}_{\log },\psi )$ where, among other things, we proved that the theory $T_{\log }=\mathrm{Th}({\mathrm{\Gamma }}_{\log },\psi )$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether $T_{\log }$ has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: $T_{\log }$ does have NIP. Our method of proof relies on a complete survey of the $1$-types of $T_{\log }$, which, in the presence of QE, is equivalent to a characterization of all simple extensions $\mathrm{\Gamma }⟨\alpha ⟩$ of $\mathrm{\Gamma }$. We also show that $T_{\log }$ does not have the Steinitz exchange property and we weigh in on the relationship between models of $T_{\log }$ and the so-called \emph{precontraction groups} of~\cite{kuhlmann1}.