Constructive symbolic presentations of rank one measure-preserving systems

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.