Model companion of ordered theories with an automorphism

Kikyo and Shelah showed that if $T$ is a theory with the Strict Order Property in some first-order language $\mathcal{L}$, then in the expanded language ${\mathcal{L}}_{\sigma }:=\mathcal{L}\cup \{\sigma \}$ with a new unary function symbol $\sigma$, the bigger theory $T_\sigma := T\cup{\sigma \mbox{is an} \mathcal{L}\mbox{-automorphism''}\}$ does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory $T_\sigma$ as the base theory $T$ together with a restricted’’ class of automorphisms, then $T_{\sigma }$ can have a model companion in ${\mathcal{L}}_{\sigma }$. We show this in the context of linear orders and ordered abelian groups.