The Parametrisation and Element of Volume of the Unitary Symplectic Group.

An essential component of the proof is the following lemma. LEMMA C. Given u subharmonic on F and f locally of type-Bi, let U be defined on G as the upper function of sup u(p). If U < + c, then U is f(p) =a subharmonic on G. A consequence of theorem B is that the classical theorem of Iversen may be extended to maps which are locally of type-Bl. Lemma C admits a number of applications to other problems. By way of illustration we cite: THEOREM D: Let G have positive ideal boundary and let u be singular positive harmonic on G. If f:F -G is of type-BI, then u of is singular on F. Theorem D leads to the following consequence for convergent infinite Blaschke products: the set of Fatou boundary values of modulus one of a convergent infinite Blaschke product is the unit circumference and each such Fatou boundary value of modulus one is attained at infinitely many points of the unit circumference.