Abstract: Two finite words u and v are called abelian equivalent if each letter of the alphabet occurs the same number of times in both u and v. The abelian subshift Ax of an infinite word x is the set of words y such that, for each factor u of y, there exists a factor v of x such that u and v are abelian equivalent. The notion of an abelian subshift gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which Ax equals the shift orbit closure Ωx. On the other hand, the abelian subshift of the Thue-Morse word contains uncountably many minimal subshifts. In this talk we discuss general properties of abelian subshifts. In particular, we consider the abelian subshifts of binary words, non-binary balanced words, and characterize abelian subshifts of aperiodic words of minimal complexity over an alphabet of cardinality k for each k ≥ 2.

Cite: Puzynina S.
Abelian subshifts
Joint Mathematics Meeting 15-18 Jan 2020