## Constructive symbolic presentations of rank one measure-preserving systems

### Volume 150 / 2017

Colloquium Mathematicum 150 (2017), 243-255
MSC: Primary 37A05, 37B10, 28D05.
DOI: 10.4064/cm7124-3-2017
Published online: 10 August 2017

#### Abstract

Given a rank one measure-preserving system defined by cutting and stacking with spacers, we produce a rank one binary sequence such that its orbit closure under the shift transformation, with its unique nonatomic invariant probability, is isomorphic to the given system. In particular, the classical dyadic odometer is presented in terms of a recursive sequence of blocks on the two-symbol alphabet $\{0,1\}$. The construction is accomplished using a definition of rank one in the setting of adic, or Bratteli–Vershik, systems.