On the subsystems of certain sofic shifts

TL;DR

This paper extends a classical embedding theorem from subshifts of finite type to certain classes of sofic shifts, clarifying conditions under which one shift can be embedded into another.

Contribution

It generalizes Krieger's theorem on embeddings from finite type shifts to specific sofic shifts, broadening the applicability of the original result.

Findings

The necessary conditions for embedding are also sufficient for certain sofic shifts.

The extension applies to classes of sofic shifts beyond finite type shifts.

The result clarifies the relationship between periodic points and embeddings in symbolic dynamics.

Abstract

For an aperiodic subshift of finite type $Y$ and for a subshift $X$ with topological entropy less than the topological entropy of $Y$, a theorem is proved in Krieger: On the subsystems of topological Markov chains, Ergodic Theory \& dynamical systems 1982 $\bold{2}$, 195-202, that says that the necessary condition on the periodic points of $X$ and $Y$ for the existence of an embedding of $X$ into $Y$ is also sufficient for the existence of an embedding of $X$ into $Y$. In this note we point out that this theorem extends to certain classes of sofic shifts as target shifts.