Summary: 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 \(^\omega uvu^\omega\), 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 \(^\omega uvu^\omega\) are precisely the biinfinite words \({\mathbf w}=\cdots a_{-2}a_{-1}a_0a_1a_2\cdots\) for which there exists a pair \((\ell_0,r_0)\) of integers with \(\ell_0<r_0\) such that, for every integers \(\ell\leq \ell_0\) and \(r\geq r_0\), the factor \(a_\ell\cdots a_{\ell_0} \cdots a_{r_0}\cdots a_r\) is a bordered word. The words of the form \(^\omega uvu^\omega\) are also characterized as being those biinfinite words \({\mathbf w}\) that admit a left recurrent unbordered factor (i.e., an unbordered factor of \({\mathbf w}\) that has an infinite number of occurrences “to the left” in \({\mathbf w})\) 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.