Biinfinite words with maximal recurrent unbordered factors

A finite non-empty word z is said to be a border of a finite non-empty word w if w = uz = zv for some non-empty words u and v. A finite non-empty word is said to be bordered if it admits a border, and it is said to be unbordered otherwise. In this paper, we give two characterizations of the biinfinite words of the form uvu, where u and v are finite words, in terms of its unbordered factors. The main result of the paper states that the words of the form uvu are precisely the biinfinite words w = · · · a−2a−1a0a1a2 · · · for which there exists a pair (l0, r0) of integers with l0 < r0 such that, for every integers l ≤ l0 and r ≥ r0, the factor al · · · al0 · · · ar0 · · · ar is a bordered word. The words of the form uvu are also characterized as being those biinfinite words w that admit a left recurrent unbordered factor (i.e., an unbordered factor of w that has an infinite number of occurrences “to the left” in w) of maximal length that is also a right recurrent unbordered factor of maximal length. This last result is a biinfinite analogue of a result known for infinite words.