On the value group of exponential and differential ordered fields

The first chapter comprises a survey of valuations on totally ordered structures, developing notation and properties. A contraction map is induced by the exponential map on the value group $G$ of an ordered exponential field $K$ with respect to the natural valuation $v_{G}$. By studying the algebraic properties of Abelian groups with contractions, the theory of these groups is shown to be model complete, complete, decidable and to admit elimination of quantifiers. Hardy fields provide an example of non-archimedean exponential fields and of differential fields and therefore, they play a very important role in our research.In accordance with Rosenlicht we define asymptotic couples and then give a short exposition of some basic facts about asymptotic couples. The theory $T_{P}$ of closed asymptotic triples, as defined in Section 2.4, is shown to be complete, decidable and to have elimination of quantifiers. This theory, as well as the theory $T$ of closed $H$-asymptotic couples do not have the independence property. The main result of the second chapter is that there is a formal connection between asymptotic couples of $H$-type and contraction groups.A given valuation of a differential field of characteristic zero is a differential valuation if an analogue of l'Hospital's rule holds. We present in the third chapter, a survey of the most important properties of a differential valuation. The theorem of M. Rosenlicht regarding the construction of a differential field with given value group is given with a detailed proof. There exists a Hardy field, whose value group is a given asymptotic couple of Hardy type, of finite rank. We also investigate the problem of asymptotic integration