Linearly invariant families of holomorphic mappings of a ball. The dimension reduction method

The notion of a linearly invariant family of mappings of a ball in $C$ was introduced in the article "Pfaltzgraff J. A., Distortion of locally biholomorphic maps of the $n$-ball, Complex Variables, 33, 239–253 (1997)" it generalizes the classical case $n=1$ studied earlier by Ch. Pommerenke and other authors. In the indicated article, Pfaltzgraff in particular obtained and used a false equality (5.3). Application of this equality also underlies some assertions in other articles. Consequently, some theorems remain unproven. We propose the dimension reduction method which enables us to save the proof and obtain new results on linearly invariant families of mappings of a ball. The idea of the method is simple and consists in reduction of a problem posed for linearly invariant families in $C^n$ to a problem for the classical case of a disk $(n=1)$.