Transition Property for $\alpha$-Power Free Languages with $\alpha \ge 2$ and $k\ge 3$ Letters

In 1985, Restivo and Salemi presented a list of five problems concerning power free languages. Problem $4$ states: Given $\alpha$-power-free words $u$ and $v$, decide whether there is a transition from $u$ to $v$. Problem $5$ states: Given $\alpha$-power-free words $u$ and $v$, find a transition word $w$, if it exists. Let ${\mathrm{\Sigma }}_k$ denote an alphabet with $k$ letters. Let $L_{k,\alpha }$ denote the $\alpha$-power free language over the alphabet ${\mathrm{\Sigma }}_k$, where $\alpha$ is a rational number or a rational “number with $+$”. If $\alpha$ is a “number with $+$” then suppose $k\ge 3$ and $\alpha \ge 2$. If $\alpha$ is “only” a number then suppose $k=3$ and $\alpha >2$ or $k>3$ and $\alpha \ge 2$. We show that: If $u\in L_{k,\alpha }$ is a right extendable word in $L_{k,\alpha }$ and $v\in L_{k,\alpha }$ is a left extendable word in $L_{k,\alpha }$ then there is a (transition) word $w$ such that $uwv\in L_{k,\alpha }$. We also show a construction of the word $w$.