Polynomial convexity of the closure of bounded pseudoconvex domains and its applications in dense holomorphic curves

In this paper, we prove that the closure of a $C^2$-smooth bounded strongly pseudoconvex domain is polynomially convex if it is invariant under positive time flows of a holomorphic vector field that has a globally attracting fixed point inside the domain. We also provide a sufficient condition for a bounded pseudoconvex domain so that its closure is polynomially convex. We show that if a class of bounded pseudoconvex domain Ω in $C^n$ which are invariant under the positive time flow of certain complete holomorphic vector fields, then given any connected complex manifold Y, there exists a holomorphic map from the unit disc to the space of all holomorphic maps from Ω to Y whose image is dense in $O(\Omega ,Y)$. This also yields us the existence of a $O(\Omega ,Y)$-universal map for any generalized translation on Ω, which implies the hypercyclicity of certain composition operators on $O(\Omega ,Y)$.