Encoding subshifts through sliding block codes

TL;DR

This paper generalizes Krieger's embedding theorem for subshifts, providing conditions under which a lower entropy subshift can be embedded into a mixing shift while preserving injectivity through a sliding block code.

Contribution

It extends Krieger's embedding theorem to a broader class of subshifts, linking topological entropy and sliding block codes in a new way.

Findings

Established necessary and sufficient conditions for embeddings of subshifts with lower entropy.

Generalized Krieger's embedding theorem in the context of zero-error information theory.

Provided a framework for embedding subshifts via sliding block codes with entropy constraints.

Abstract

We prove a generalization of Krieger's embedding theorem, in the spirit of zero-error information theory. Specifically, given a mixing shift of finite type $X$, a mixing sofic shift $Y$, and a surjective sliding block code $\pi: X \to Y$, we give necessary and sufficient conditions for a subshift $Z$ of topological entropy strictly lower than that of $Y$ to admit an embedding $\psi: Z \to X$ such that $\pi \circ \psi$ is injective.