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The full Kolakoski shift is topologically transitive
Abstract
We prove that the full Kolakoski shift---namely, the subshift of all smooth sequences over the alphabet {1,2}---is topologically transitive. Consequently, we demonstrate the existence of a smooth sequence whose factors are invariant under both reversal and bitwise complementation. As a further application, we prove the existence of arbitrarily long smooth words containing an equal number of occurrences of the symbols $1$ and $2$, providing evidence in support of Keane's conjecture. Our approach leverages the hierarchical structure of the full Kolakoski shift to reimagine the shift action in terms of a self-similar semigroup, which allows techniques from group theory and combinatorics to be applied.
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BibTeX
@misc{mitchell2026full,
title = {{The full Kolakoski shift is topologically transitive}},
author = {Andrew C. Mitchell and James J. Walton and Michael F. Whittaker},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06878},
eprint = {2611.06878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06878},
}
RIS
TY - UNPB
TI - The full Kolakoski shift is topologically transitive
AU - Andrew C. Mitchell
AU - James J. Walton
AU - Michael F. Whittaker
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06878
UR - https://arxiv.org/abs/2611.06878
AB - We prove that the full Kolakoski shift---namely, the subshift of all smooth sequences over the alphabet {1,2}---is topologically transitive. Consequently, we demonstrate the existence of a smooth sequence whose factors are invariant under both reversal and bitwise complementation. As a further application, we prove the existence of arbitrarily long smooth words containing an equal number of occurrences of the symbols $1$ and $2$, providing evidence in support of Keane's conjecture. Our approach leverages the hierarchical structure of the full Kolakoski shift to reimagine the shift action in terms of a self-similar semigroup, which allows techniques from group theory and combinatorics to be applied.
ER -