A characterization of periodicity of bi-infinite words

A finite word is called bordered if it has a proper prefix which is also a suffix of that word. Costa proves in [Theoret. Comput. Sci. 290(3) (2003) 2053-2061] that a bi-infinite word w is of the form fgfωω, for some finite words f and g, if, and only if, there is a factorization w=suv, with u∈A* such that every factor s′uv′, with s′≼s and v′⩽v, is bordered. We present a shorter proof of that result in this paper.