Symmetries and regularity for holomorphic maps between balls

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