Restivo Salemi property for $\alpha$-power free languages with $\alpha \ge 5$ and $k\ge 3$ letters

Abstract

In 2009, Shur published the following conjecture: Let $L$ be a power-free language and let $e(L)\subseteq L$ be the set of words of $L$ that can be extended to a bi-infinite word respecting the given power-freeness. If $u,v\in e(L)$ then $uwv\in e(L)$ for some word $w$. Let $L_{k,\alpha }$ denote an $\alpha$-power free language over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and $k$ is positive integer. We prove the conjecture for the languages $L_{k,\alpha }$, where $\alpha \ge 5$ and $k\ge 3$.

arXiv admin note: substantial text overlap with arXiv:2202.12038