  We describe the action of the symplectic group on the homogeneous space of
squeezed states (quantum blobs) and extend this action to the semigroup. We
then extend the metaplectic representation to the metaplectic (or oscillator)
semigroup and study the properties of such an extension using Bargmann-Fock
space. The shape geometry of squeezing is analyzed and noncommuting elements
from the symplectic semigroup are proposed to be used in simultaneous
monitoring of noncommuting quantum variables - which should lead to fractal
patterns on the manifold of squeezed states.
