Fibonacci word patterns in two-way infinite Fibonacci words

The studies of 1-Fibonacci word patterns and 0-Fibonacci word patterns were initiated by Turner (1988) [18], and Chuan (1993) [2] respectively. It is known that each proper suffix of the infinite Fibonacci word is an r-Fibonacci word pattern, r∈{0,1}. In this paper, we consider the suffixes of the two two-way infinite extensions G and G′ of the infinite Fibonacci word. We obtain necessary and sufficient conditions for suffixes of G and G′ to be r-Fibonacci word patterns. This gives us all the mechanical words with slope which are r-Fibonacci word patterns. All possible r-seed words of each of them are determined. Finally, images of suffixes of G and G′ under the action of certain Sturmian morphisms are computed.