A simple representation of subwords of the Fibonacci word

We introduce a representation of subwords of the infinite Fibonacci word f∞ by a specific concatenation of finite Fibonacci words. It is unique and easily computable by backward processing of the given word. This provides an efficient recognition algorithm for subwords of f∞ as well as a full description of their occurrences in f∞. Our representation yields a natural notion of rank of a subword, which explains main properties of subwords of f∞ in a unified way.