Automorphisms of valued fields: amalgamation and existential closedness

A valued difference field is a field equipped with a valuation and an automorphism which preserves the valuation ring setwise. In this talk I will discuss various results on these objects, concerning the existence of a section of the valuation compatible with the automorphism and of extensions of the valuation to difference fields extension, the solvability of amalgamation problems of valued difference fields, and the classification of the theory of valued difference fields in the sense of positive model theory. This is joint work with Jan Dobrowolski and Rosario Mennuni.