For a semi-group Γ
of positive linear contractions on L1 of a σ-finite measure space (X,𝒜,μ),
strongly continuous on (0,∞), there are two ratio ergodic theorems: one, due to
Chacon and Ornstein, describes the behavior at infinity; the other one, due to
Krengel-Ornstein-Akcoglu-Chacon, describes the “local” behavior. In the present
paper we attempt to generalize these results to the case when the semigroup
is only uniformly bounded. Then the space X decomposes into two parts,
Y and Z, called the remaining and the disappearing part, and both ratio
theorems are shown to hold on Y . The ratio theorem at infinity fails on
Z.
