Sur une généralisation de la notion de système dynamique de rang un définie par une propriété de pistage (On a weak version of the rank one property defined by shadowing)

Jerome Buzzi (1)

1 IML - Institut de mathématiques de Luminy

We investigate a shadowing property which appears naturally in the study of piecewise monotonic maps of the interval. It turns out to be a weak form of the rank one property, a well-known notion in abstract ergodic theory. We show that this new property is implied by finite or even local rank, but that it is logically independent of loose Bernoulliness. We give (counter)examples, including L.B.\ systems with arbitrarily high-order polynomial complexity. The shadowing property defines a small subset of all zero-entropy systems, in the sense 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 the link between the shadowed system and the sequence assumed by the shadowing property.

Mots clés: shadowing; rank one; local rank; loosely Bernoulli; zero entropy; measure-theoretic entropy

Domaines: Systèmes dynamiques [math.DS]

Jerome Buzzi. Sur une généralisation de la notion de système dynamique de rang un définie par une propriété de pistage (On a weak version of the rank one property defined by shadowing). 1998. ⟨hal-00171713⟩