Construction of a Bi-infinite Power Free Word with a Given Factor and a Non-recurrent Letter

Let $L_{k,\alpha}^{\mathbb{Z}}$ denote the set of all bi-infinite $\alpha$-power free words over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and integer. We prove that if $\alpha\geq 5$, $k\geq 3$, $v\in L_{k,\alpha}^{\mathbb{Z}}$, $w$ finite factor $v$, then there are $\widetilde v\in L_{k,\alpha}^{\mathbb{Z}}$ letter $x$ such v$ has only finitely many occurrences in v$.