We propose a model complete theory of valued fields equipped with an $\omega$ increasing endomorphism which is not onto. The key to the quantifier elimination statement is a study of generic types and a systematic use of descent. As a consequence, we derive the decidability of the elementary theory of the Frobenius action on separably algebraically closed valued fields, generalizing results of Chatzidakis and Hrushovski. This is joint work with Yatir Halevi.[-]