Polynomial shape adic systems are inherently expansive

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.