On bispecial factors of the Thue–Morse word

Abstract

A factor f of the Thue–Morse word t in a two-letter alphabet { a, b} is bispecial if fa, fb, af, bf are still factors. We calculate the enumeration function β giving for each integer n > 0 the number β( n) of bispecial factors of length n. We prove that β takes only the values 0 and 2, and that β( n) = 2 iff n = 2 k or n = 3.2 k , with k ⩾ 0. A simple construction of the bispecial factors from the prefixes of t is given. We consider, moreover, strictly bispecial factors of t, i.e. factors f such that afa, afb, bfa, bfb are still factors of t . We prove that these factors are precisely the bispecial factors of t whose length is n = 2 k , k > 0.