Polynomial shape adic systems are inherently expansive

TL;DR

This paper demonstrates that polynomial shape adic systems, a class of Bratteli-Vershik systems defined by multivariable polynomials, can be essentially faithfully encoded with minimal loss, highlighting their expansiveness.

Contribution

It establishes that polynomial shape adic systems are inherently expansive and can be encoded injectively with negligible exceptions, advancing understanding of their dynamical properties.

Findings

Polynomial shape adic systems are inherently expansive.

They can be encoded faithfully with negligible exceptions.

Encoding is injective off a negligible set for all edge orderings.

Abstract

To study any dynamical system it is useful to find a partition that allows essentially faithful encoding (injective, up to a small exceptional set) into a subshift. Most topological and measure-theoretic systems can be represented by Bratteli-Vershik (or adic, or BV) systems. So it is natural to ask when can a BV system be encoded essentially faithfully. We show here that for BV diagrams defined by homogeneous positive integer multivariable polynomials, and a wide family of their generalizations, which we call polynomial shape diagrams, for every choice of the edge ordering the coding according to initial path segments of a fixed finite length is injective off of a negligible exceptional set.