Embedding asymptotically expansive system

TL;DR

This paper establishes a Krieger-like embedding theorem for asymptotically expansive systems with the small boundary property, linking them to full shifts and exploring measure-theoretic embeddings and extensions.

Contribution

It introduces a new embedding theorem for asymptotically expansive systems, connecting them to full shifts and analyzing measure-theoretic properties.

Findings

Embedding in K-full shift with topological entropy less than log K

Invariant measure map is a topological embedding

Inverse embedding extends to a faithful symbolic extension

Abstract

We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system $(X; T)$ embeds in the $K$-full shift with $h_{top}(T) < \log K $ and $\sharp Per_n(X; T) \leq \sharp Per_n(\{1,...,K\}^{\mathbb{Z}};\sigma)$ for any integer $n$. The embedding is in general not continuous (unless the system is expansive and $X$ is zero-dimensional) but the induced map on the set of invariant measures is a topological embedding. It is shown that this property implies asymptotical expansiveness. We prove also that the inverse of the embedding map may be continuously extended to a faithful principal symbolic extension.