Transition Property for $α$-Power Free Languages with $α\geq 2$ and $k\geq 3$ Letters
英文摘要：
In 1985, Restivo and Salemi presented a list of five problems concerning power free languages. Problem $4$ states: Given $α$-power-free words $u$ and $v$, decide whether there is a transition from $u$ to $v$. Problem $5$ states: Given $α$-power-free words $u$ and $v$, find a transition word $w$, if it exists. Let $Σ_k$ denote an alphabet with $k$ letters. Let $L_{k,α}$ denote the $α$-power free language over the alphabet $Σ_k$, where $α$ is a rational number or a rational "number with $+$". If $α$ is a "number with $+$" then suppose $k\geq 3$ and $α\geq 2$. If $α$ is "only" a number then suppose $k=3$ and $α>2$ or $k>3$ and $α\geq 2$. We show that: If $u\in L_{k,α}$ is a right extendable word in $L_{k,α}$ and $v\in L_{k,α}$ is a left extendable word in $L_{k,α}$ then there is a (transition) word $w$ such that $uwv\in L_{k,α}$. We also show a construction of the word $w$.
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