bibkey: kang2026lifts
authors: Z. Kang; T. Jäger
year: 2026
title: "On lifts of strictly ergodic subshifts by permutative sliding block codes"
doi: 10.48550/arXiv.2609.01140
url: https://arxiv.org/abs/2609.01140v1
claim: "Question 1.10 asks whether a minimal or strictly ergodic subshift admits a permutative code of length at least two with a minimal or strictly ergodic complete inverse image, optionally under a Toeplitz hypothesis."
strata_touched:
  - D5/S1/Dynamics/Symbolic/PermutativeLifts/UniversalRefutation
license: arXiv.org perpetual non-exclusive license
triage: anchor

Permutative lifts and the general minimal branch

The source’s Question 1.10 asks: given a minimal/strictly ergodic (Toeplitz) subshift, does some permutative sliding block code of length at least two have a minimal/strictly ergodic complete inverse image? Permutativity means bijectivity in each endpoint with every other letter fixed. The same finite alphabet is used for the source and image of the code. Theorem 1.3 describes the entire lift as a finite permutation skew product with the actual prefix fibre, of cardinality κ^(ℓ−1).

The linked formal construction addresses the general minimal branch without the optional Toeplitz restriction. Its single infinite binary minimal base has nonminimal entire lifts for all lengths and all such codes. It does not claim the Toeplitz branch or a formal unique-ergodicity strengthening.

The coboundary mechanism is prior work, including Berthé, Goulet-Ouellet, Nyberg-Brodda, Perrin and Petersen, arXiv:2403.17892v3, Definition 7.3, Proposition 7.4 and Theorem 7.5. The candidate contribution is the explicit recognizable factorial base simultaneously neutralizing all finite letter actions. A bounded primary-source search on 2026-10-31 did not locate this same uniform construction; it establishes neither worldwide absence nor priority. Cited works announced as in preparation were unavailable.

Verified locator

DOI: 10.48550/arXiv.2609.01140. Version URL: https://arxiv.org/abs/2609.01140v1. Question: https://arxiv.org/html/2609.01140v1#S1.Thmdefinition10. The retrieved version is dated 2026-09-01 and states the arXiv.org perpetual non-exclusive license. This note cites mathematical statements and does not redistribute the paper’s text or code.