  We study the theory of Banach $L^p$ lattices with a distinguished
automorphism, in the framework of continuous logic. Using a functional version
of the Rokhlin lemma, we prove that it admits a model companion, which is
stable and has quantifier elimination. We show that the types of this theory
that are not trivial cannot be isolated. We then use this result to obtain a
proof of the absence of comeagre conjugacy classes in
$\operatorname{Aut}^*({\mu})$, the Polish group of non-singular transformations
of a standard probability space.
