Title: On the subsystems of a sofic shift
Abstract:
Krieger's embedding theorem provides simple necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing subshift of finite type:
For strict embedding, the only obstructions are entropy and counting period points.
Soon after it was  asked  how this result extends to sofic shifts.
It was quickly realized that the statement of Krieger's theorem cannot be naively extended to sofic shifts.
Nearly half a century later, it remains an open problem to find explicit  necessary and sufficient conditions for embedding an arbitrary subshift into a given topologically mixing sofic shift.
A number of key results obtained over the years shed light on the problem, possibly hinting on the nature of the solution one should expect:
In a 1983 paper Boyle proved a theorem that provides sufficient conditions  for an arbitrary subshift to embed in a given topologically mixing sofic shift.
In 2004  Klaus Thomsen obtained significant progress on this problem in the early 21st century, by finding necessary and sufficient conditions for an irreducible SFT to embed in a given  topologically mixing, revealing a surprising ``decomposition'' of sofic shifts, that can be non-trivial even in the mixing case.
In 2022 Sophie Morrian (formerly MacDonald) obtained a generalization of Krieger's embedding theorem that provides sufficient conditions for an arbitrary subshift to embed in a given topologically mixing SFT, in such a way that the embedding remains injective after applying  a given factor map.
I plan to discuss the above historical results, along with some very recent ones, based on  ongoing joint work with Brian Marcus, Klaus Thomsen and  Chengyu Wu.