Countless Contractions

We examine the model-theoretic properties of the induced structure on the archimedean classes of a non-archimedean expansion of $\mathbb{R}_{\exp}$ equipped with trans-exponential functions. This is mainly done by studying Contraction groups and Asymptotic triples.

Contraction groups are a type of model-theoretic structure introduced by F.V Kuhlmann consisting of an ordered abelian group $G$ along with a unary function $\chi : G \to G$ which collapses entire archimedean classes to a single point. The canonical example being the action of the logarithm on the archimedean classes of a non-archimedean model of $\mathbb{R}_{\exp}$. In the papers \cite{FVKuhlmann1994AbelianGWContractionsQEProof} and \cite{FVKuhlmann1995AbelianGWContractionsWO-MinProof}, it was shown that the theory of centripetal contraction groups has quantifier elimination and is weakly o-minimal.

Similarly, Asymptotic triples consist of an ordered abelian group $G$ and a unary function $\psi: G^{\not = 0} \to G$ collapsing entire archimedean classes to a point, with the canonical example being the action of the \textbf{logarithmic derivative} on the archimedean classes of the germs of functions on $\mathbb{R}_{\exp}$. In the works of Aschenbrenner and Van den Dries it was also shown that some formulation of these into first-order logic has quantifier elimination and is weakly o-minimal.

The works mentioned above only deal with regular logarithms and exponentials. In this thesis, we extend the works above, to so-called `Hyper-logarithms' and `Trans-exponentials', which can intuitively be thought of as the composition of $\log$ and $\exp$ infinitely many times. The main results for this thesis are quantifier elimination and weak o-minimality for $n$-contraction groups along with quantifier elimination for $\theta-L$-contraction groups.