Existence of minimal or strictly ergodic permutative lifts for a given Toeplitz subshift

Determine whether every minimal or strictly ergodic Toeplitz subshift over the alphabet \(\mathcal A_\kappa\) admits a permutative sliding block code \(h\in\mathcal F(\kappa,\ell)\) of some length \(\ell\geq 2\) such that the lift \(X=\eta_h^{-1}(Y)\) is, respectively, minimal or strictly ergodic.

Background

The paper studies finite-to-one extensions of symbolic dynamical systems obtained by taking the preimage of a subshift under a permutative sliding block code. A central issue is determining when minimality and unique ergodicity of the base subshift are preserved by this lifting procedure.

The authors construct examples showing that, for Toeplitz subshifts, a fixed permutative block-code length can support minimal or strictly ergodic lifts, while other choices can produce non-minimal or minimal but non-uniquely ergodic lifts. They therefore leave unresolved the converse-type existence question of whether an arbitrary minimal or strictly ergodic Toeplitz subshift admits at least one suitable permutative lift with the corresponding dynamical property.

References

One question which we leave open here is the following: Given a minimal/strictly ergodic (Toeplitz) subshift $Y\ssq\cA_\kappa\Z$, does there exist a permutative sliding block code $h\in\cF(\kappa,\ell)$ of length $\ell\geq 2$ such that $X=\eta_h{-1}(Y)$ is minimal/strictly ergodic?

— Lifts of strictly ergodic subshifts by permutative sliding block codes  (2609.01140 - Kang et al., 1 Sep 2026) in Question in Section 1, immediately following Theorem 1.12 (Introduction)