The linear fractional model on the ball

Given a holomorphic self-map φ of the ball of CN , we study whether there exists a map σ and a linear fractional transformation A such that σ ◦ φ = A ◦ σ. This is an important result when N = 1 with a great number of applications. We extend this result to the multi-dimensional setting for a large class of maps. Applications to commuting holomorphic self-maps are given.