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)

TL;DR

This paper explores a shadowing property in piecewise monotonic interval maps, revealing it as a weak form of the rank one property, with implications for understanding zero-entropy systems and their complexity.

Contribution

It introduces a new shadowing property related to rank one, analyzes its implications, and provides examples showing its independence from loose Bernoulliness.

Findings

Shadowing property is implied by finite or local rank.

It is independent of loose Bernoulliness.

The property characterizes a small subset of zero-entropy systems.

Abstract

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.