Embedding under a preimage bound

Definitions and mathematical statement

Let $\mathcal A$ be the finite alphabet of a one-sided subshift $S\subseteq\mathcal A^{{\mathbb{N}}_0}$, equipped with the product topology and the left shift $\sigma_S$. Thus $S$ is closed and shift invariant; whether $\sigma_S$ must be onto is not made explicit in the source. Let \[ T=\{1,\ldots,N\}^{{\mathbb{N}}_0} \] be the full one-sided shift with shift $\sigma_T$. Here ${\mathbb{N}}_0=\{0,1,2,\ldots\}$ and $N$ is a positive integer. Write $h(S)$ for topological entropy, equivalently the exponential growth rate of the numbers of allowed words.

Under the supplied hypotheses \[ h(S)<\log N,\qquad \forall x\in S,\quad |\{y\in S:\sigma_S(y)=x\}|\le N, \] must there exist a continuous injection $\iota:S\to T$ with \[ \iota\circ\sigma_S=\sigma_T\circ\iota\ ? \] Because $S$ is compact and $T$ is Hausdorff, such an injection is a homeomorphism onto its image, and that image is a subshift. This is the topological meaning of "isomorphic to a subshift of $T$" used here, rather than a merely measurable isomorphism.

The assertion is retained exactly with its supplied hypotheses. In particular the source provides no separate periodic-point capacity condition, although an embedding into $T$ must inject each set of points fixed by $\sigma_S^k$ into the $N^k$ points fixed by $\sigma_T^k$. Whether a necessary periodic-orbit hypothesis or another restriction has been lost in extraction requires review; it is not silently added.

Short English statement

Does entropy strictly below that of the full $N$-symbol one-sided shift, together with at most $N$ immediate predecessors per point, force a topological embedding into that full shift? The local preimage bound and the original hypotheses are preserved, with missing periodic-point restrictions explicitly flagged.

The supplied research summary calls the assertion open and plausibly true, and reports finding no resolution through 2026. It supplies no theorem or source check resolving the formulation issue. This remains an OPEN-TRIAGE assessment; provenance records no later authoritative resolution check of the 2008 source.

Sources

• Local UnsolvedMath registry, permanent ID 4600032, source code catalogue entry, "Embedding under a preimage bound," proposed year recorded as 2008; no proposer is specified. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Mike Boyle, Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 20.2, https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf; pdf-text extraction, accessed 2026-07-29. The research summary describes the question as posed by Boyle.
• The supplied summary contains no later primary reference supporting its open-status assessment or verifying completeness of the extracted hypotheses.
• Source provenance marks status evidence NEEDS_REVIEW and rights NEEDS_REVIEW: the source presents the item as open, no later authoritative resolution check is recorded, and redistribution permission was not established despite verified public access. This qualification is separate from the dataset licence.