Given a field $K$, one can enrich it via a valuation, an automorphism or both structures at the same time in a compatible way. In all of these three cases, the model theory is well-understood (under some further hypotheses, e.g. henselianity of the valuation or the automorphism being isometric). I'd like to explore the connection between these different, but related, structures and one natural question that came up is the following: given an automorphism $\sigma$ of $K$, can one find a non-trivial valuation $v$ on $K$ such that $\sigma$ is isometric for $v$, i.e. such that $v(\sigma(x)) = v(x)$ for every $x$? For example, if $K$ has characteristic $p>0$, then such a valuation cannot exist for $x \mapsto x^p$.
(This seems to be equivalent to finding a fixed point for automorphisms of the valuation spectrum of $K$, but I don't know enough about valuation spectra to be sure of this.)
