  We extend the notion of the canonical extension of automorphisms of type III
factors to the case of endomorphisms with finite statistical dimensions.
Following the automorphism case, we introduce two notions for endomorphisms of
type III factors: modular endomorphisms and Connes-Takesaki modules. Several
applications to compact groups of automorphisms and subfactors of type III
factors are given from the viewpoint of ergodic theory.
