An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds

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.