As models of strictly pseudoconvex domains, we consider holomorphic functions on the unit ball \ball{n}=\{z\in\C^n:|z|<1\}. In particular, we focus on proper holomorphic maps \ball{n}\to\ball{N}. In the equidimensional case N=n, proper holomorphic maps are automorphisms. We discuss the parameters associated to automorphisms, and more generally involutions and their higher-order analogues.We then define the mixed spaces \ball{n,k}=\{(z,s)\in\C^n\times\R^k:|z|^2+|s|^2<1\}, and address similar questions regarding proper maps, automorphisms, and involutions in the new setting. In particular, we show how to recover the parameters that determine an automorphism of \ball{n,k} using the germ at z=0. We also specify necessary conditions on involutions in both the \ball{n} and \ball{n,k} settings
