Squeeze Mapping - Group Theory

Group Theory

If r and s are positive real numbers, the composition of their squeeze mappings is the squeeze mapping of their product. Therefore the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicative group of positive real numbers. An additive view of this group arises from consideration of hyperbolic sectors and hyperbolic angles. In fact, the invariant measure of this group is hyperbolic angle.

From the point of view of the classical groups, the group of squeeze mappings is SO+(1,1), the identity component of the indefinite orthogonal group of 2 × 2 real matrices preserving the quadratic form This is equivalent to preserving the form via the change of basis and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group SO(2) (the connected component of the definite orthogonal group) preserving quadratic form ) as being circular rotations.

Note that the "SO+" notation corresponds to the fact that the reflections and are not allowed, though they preserve the form (in terms of x and y these are and ); the additional "+" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity component because the group O(1,1) has 4 connected components, while the group O(2) has 2 components: SO(1,1) has 2 components, while SO(2) only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups – in this case SO(1,1) ⊂ SL(2) – of the subgroup of hyperbolic rotations in the special linear group of transforms preserving area and orientation (a volume form). In the language of Möbius transforms, the squeeze transformations are the hyperbolic elements in the classification of elements.