Constructive symbolic presentations of rank one measure-preserving systems

TL;DR

This paper presents a method to construct symbolic representations of rank one measure-preserving systems using recursive sequences, enabling explicit isomorphisms with classical systems like the dyadic odometer.

Contribution

It introduces a new construction of symbolic models for rank one systems via adic and Bratteli-Vershik frameworks, providing explicit recursive sequences.

Findings

Constructs symbolic models for rank one systems

Shows isomorphism with classical systems like the dyadic odometer

Uses adic and Bratteli-Vershik systems for construction

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.