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.