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We investigate a shadowing property which appears naturally in the study ofpiecewise monotonic maps of the interval. It turns out to be a weak form of therank one property, a well-known notion in abstract ergodic theory. We show thatthis new property is implied by finite or even local rank, but that it islogically independent of loose Bernoulliness. We give (counter)examples,including L.B. systems with arbitrarily high-order polynomial complexity. Theshadowing property defines a small subset of all zero-entropy systems, in thesense that it defines a closed set with empty interior with respect to the$\db$-metric, induced by the Hamming distance. We also make some remarks on thelink between the shadowed system and the sequence assumed by the shadowingproperty.

Sur une g\'en\'eralisation de la notion de syst\`eme dynamique de rang  un d\'efinie par une propri\'et\'e de pistage (On a weak version of the rank  one property defined by shadowing)