Construction of a bi-infinite power free word with a given factor and a non-recurrent letter

Abstract

In 2009, Shur published the following conjecture: Let L be a powerfree language and let e(L) ⊆ L be the set of words of L that can be extended to a bi-infinite word respecting the given power-freeness. If u, v ∈ e(L) then uwv ∈ e(L) for some word w.Let L k,α denote an α-power free language over an alphabet with k letters, where α is a positive rational number and k is positive integer. If α ≥ 4 and k ≥ 3 then we prove that the conjecture holds for the language L k,α .