  String attractors are a combinatorial tool coming from the field of data
compression. It is a set of positions within a word which intersects an
occurrence of every factor. While one-sided infinite words admitting a finite
string attractor are eventually periodic, the situation is different for
two-sided infinite words. In this paper, we characterise the bi-infinite words
admitting a finite string attractor as the characteristic Sturmian words and
their morphic images. For words that do not admit finite string attractors, we
study the structure and properties of their infinite string attractors.
