On dominant contractions and a generalization of the zero-two law

Zaharopol proved the following result: let $T,S:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$betwopositivecontractionssuchthat$T\leq S$\mathrm{.}If$|S-T|<1$then$|S^n-T^n|<1$forall$n\in\bn$. In the present paper we generalize this result to multi-parameter contractions acting on $L^1$. As an application of that result we prove a generalization of the “zero-two” law.