Pointwise ergodic theorems for certain order preserving mappings in L1

For an order preserving map T in L1 we study the “nonlinear sums” defined recurrently by S0f = f, Sn+1f = f + T(Snf), and the “averages” Anf = Snf/(n + 1). We prove a maximal ergodic lemma if T is norm decreasing in L1, dominated estimates for M f = sup Anf when T is also L∞ nonexpansive and order continuous. If moreover T is positively homogeneous and has no fixed points ≠ 0 in Lp+ (resp. Lp) and 1 < p < ∞ then we can show that lim Anf = 0 a.e. and in Lp–norm for f ∈ Lp+ (resp. f ∈ Lp). This theorem is applied to generalized measure preserving transformations in infinite measure spaces. In this case a.e. convergence holds also for p = 1. In the finite measure case we add the assumption that T is integral preserving and that the constant functions are the only fixed points and obtain a.e. convergence of Anf to ∫ f for f ∈ L log L.