  Maps on a parameter space for expressing distribution functions are exactly
derived from the Perron-Frobenius equations for a generalized Boole transform
family. Here the generalized Boole transform family is a one-parameter family
of maps where it is defined on a subset of the real line and its probability
distribution function is the Cauchy distribution with some parameters. With
this reduction, some relations between the statistical picture and the orbital
one are shown. From the viewpoint of information geometry, the parameter space
can be identified with a statistical manifold, and then it is shown that the
derived maps can be characterized. Also, with an induced symplectic structure
from a statistical structure, symplectic and information geometric aspects of
the derived maps are discussed.
