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.