Systèmes localement de rang un. (French) Zbl 0535.28010

Ferenczi, S.

Ann. Inst. Henri Poincaré, Probab. Stat. 20, 35-51 (1984).

We say that a dynamical system is of local rank one if it can be approximated arbitrarily well by Rokhlin towers of fixed measure. We show that this property forces loosely Bernoulli (this implies the well-known result that systems of finite rank are loosely Bernoulli).
Then, we show that there are loosely Bernoulli systems which are not of local rank one: namely, systems possessing the Vershik property, like those constructed by A. Rothstein [Isr. J. Math. 36, 205-224 (1980; Zbl 0461.60054)], cannot be of local rank one.
Finally, we construct an example to show that local rank one is strictly weaker than finite rank.

MSC:

28D10	One-parameter continuous families of measure-preserving transformations
28D05	Measure-preserving transformations

Keywords:

dynamical system of local rank one; Rokhlin towers of fixed measure; loosely Bernoulli systems; Vershik property