  Consider a smooth $4$-manifold $X$ and a diffeomorphism $f : X \to X$. We
give an obstruction in the form of an adjunction inequality for an embedded
surface in $X$ to be isotopic to its image under $f$. It follows that the
minimal genus of a surface representing a given homology class and which is
isotopic to its image under $f$ is generally larger than the minimal genus
without the isotopy condition. We give examples where the inequality is strict.
We use our obstruction to construct examples of infinitely many embedded
surfaces which are all continuously isotopic but mutually non-isotopic
smoothly.
