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. Following an idea of Ornstein and Sucheston, Foguel provedthe so-called uniform “zero-two” law: let T : L 1 (X,F,µ) → L 1 (X,F,µ)be a positive contraction. If for some m ∈ N∪{0} one has kT m+1 −T m k <2, thenlim n→∞ kT n+1 − T n k = 0.There are many papers devoted to generalizations of this law. In thepresent paper we provide a multi-parametric generalization of the uniformzero-two law for L 1 -contractions. 1. IntroductionLet (X,F,µ) be a measure space with a positive σ-additive measure µ. Inwhat follows, for the sake of shortness, we denote by L 1 the usual L 1 (X,F,µ)space associated with (X,F,µ). A linear operator T : L 1 → L 1 is called apositivecontractionif Tf ≥ 0 whenever f ≥ 0 and kTk ≤ 1.Jamison and Orey [7] proved that if P is a Markov operator recurrent in thesense of Harris, with σ-nite invariant measure µ, then kP n gk 1 → 0 for everyg ∈ L 1 withRg dµ = 0 if (and only if) the chain is aperiodic. Clearly, whenthe chain is not aperiodic, taking f with positive and negative parts supportedin dierent sets of the cyclic decomposition, we have lim

ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L1-SPACES

ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L¹-SPACES