Königs maps and commutants of composition operators on the Hardy-Hilbert space of Dirichlet series

Let $\phi$ be a holomorphic map which is a symbol of a bounded composition operator $C_{\phi }$ acting on the Hardy-Hilbert space of Dirichlet series. We find a Königs map for $\phi$. We then deduce several applications on $C_{\phi }$ (e.g. on its spectrum, on its dynamical properties). In particular, we study for a large class of symbols $\phi$ if the associated composition operator has a minimal commutant.