Embedding one-sided subshifts

Definitions and mathematical statement

A one-sided subshift is a closed set $S\subset A^{\mathbb N_0}$ over a finite alphabet with $\sigma(S)\subset S$, where $(\sigma x)_n=x_{n+1}$. Its entropy is $h(S)=\lim_{n\to\infty}n^{-1}\log|\mathcal L_n(S)|$, with $\mathcal L_n(S)$ the occurring words of length $n$. Let $T\subset B^{\mathbb N_0}$ be a mixing one-sided shift of finite type: finitely many forbidden words define $T$, and for any allowed words all sufficiently long connecting lengths are possible. Assume $h(S)<h(T)$.

Give useful necessary and sufficient conditions for an embedding $\iota:S\to T$ to exist. An embedding is a continuous injective map satisfying $\iota\sigma_S=\sigma_T\iota$; compactness makes it a conjugacy from $S$ onto its image subshift. Its local coding rule may use a finite amount of future information, with no past coordinates available.

Any such map injects periodic orbits of each least period and embeds the backward preimage tree of a point of $S$ into that of its image in $T$. Thus entropy alone cannot decide the embedding question. The desired criterion must handle these compatibility restrictions throughout the one-sided system. No finite-type or sofic hypothesis is imposed on $S$, and no coding radius is specified in advance. All embeddings must preserve one shift step, rather than only an iterate or an orbit relation.

Short English statement

Find exact criteria for embedding a lower-entropy one-sided subshift into a mixing one-sided finite-type shift.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600031 (AMR-045-0031), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 20.1. Original source indexed by AMR-045-0031. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.