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. Extending what was known previously for special diagrams and particular orders, 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.