Mean ergodicity of L1 contractions and pointwise ergodic theorems

Let T be a (linear) contraction on L1 with modulus . We show that if is mean ergodic, so is T, and the averages An(T)f and An()f converge a.e. for any f ∈ L1. The mean ergodicity of is not necessary for the mean ergodicity of T together with the a.e. convergence of the averages. If {Fn} if a dominated super additive process with respect to a mean ergodic positive contraction T, then converges a.e. and in L1, to a T-invariant function.

Let T1, …, Td be commuting contractions, and let An(T1, …, Td) = An(T1)…An(Td). Then An(T1, …, Td)f converges in L1 for every f ∈ L1 if the associated Brunel operator is mean ergodic. If the Ti are positive, the converse is also true. We prove that if the moduli are mean ergodic and commute, the averages An(T1, …, Td)f and An(1, …, d)f converge a.e. for every f ∈ L1.