Symmetries and regularity for holomorphic maps between balls

Let $f:{mathbb B}^n to {mathbb B}^N$ be a holomorphic map. We study subgroups $Gamma_f subseteq {rm Aut}({mathbb B}^n)$ and $T_f subseteq {rm Aut}({mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups. When $Gamma_f$ contains the center of ${bf U}(n)$, we show that $f$ is spherically equivalent to a polynomial. When $f$ is minimal we show that there is a homomorphism $Phi:Gamma_f to T_f$ such that $f$ is equivariant with respect to $Phi$. To do so, we characterize minimality via the triviality of a third group $H_f$. We relate properties of ${rm Ker}(Phi)$ to older results on invariant proper maps between balls. When $f$ is proper but completely non-rational, we show that either both $Gamma_f$ and $T_f$ are finite or both are noncompact.