**Abstract:** The first part of the thesis concerns the existence of model companions of certain unstable theories with automorphisms. Let T be a first-order theory with the strict order property. According to Kikyo and Shelah's theorem, the theory of models of T with a generic automorphism does not have a model companion. However, existence can be restored with some restriction on the automorphism. We show the existence of model companions of the theory of linear orders with increasing automorphisms and the theory of ordered abelian groups with multiplicative automorphisms. Both these theories have the strict order property.
The second part of the thesis uses these results from the first part in the context of valued difference fields, which are valued fields with an automorphism on them. Understanding the theory of such structures requires one to specify how the valuation function interacts with the automorphism. Two special cases have been worked out before. The case of the isometric automorphism is worked out by Luc BÃ©lair, Angus Macintyre and Thomas Scanlon; the case of the contractive automorphism is worked out by Salih Azgin. These two cases, however, are two ends of a spectrum. Our goal in this thesis is to fill this gap by defining the notion of a multiplicative valued difference field. We prove an Ax-Kochen-Ershov type of result, whereby we show that the theory of such structures is essentially controlled by the theory of their so-called "residue-valuation" structures (RVs). We also prove relative quantifier elimination theorem for such structures relative to their RVs. Finally we show that in the presence of a "cross-section", we can transfer these relative completeness and relative quantifier elimination results relative to their value groups and residue fields.