ON BISPECIAL FACTORS OF THE THUE-MORSE WORD

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 beta giving for each integer n > 0 the number beta(n) of bispecial factors of length n. We prove that beta takes only the values 0 and 2 and that beta(n) = 2 iff n = 2k or n = 3 . 2k, with k greater-than-or-equal-to 0. A simple construction of the bispecial factors from the prefixes of t is given. Wc consider, moreover, strictly bispecial factors of t, i.e. factors f such that afa, afb, bfa, bfb are still factor, of t. We prove that these factors are precisely the bispecial factors of t whose length is n = 2k, k > 0.