For a rational number r such that $1<r\le 2$, an undirected r-power is a word of the form $xy{x}^{\prime }$, where the word x is nonempty, the word $x^{\prime }$ is in $\{x,x^R\}$, and we have $|xy{x}^{\prime }|/|xy|=r$. The undirected repetition threshold for k letters, denoted $\mathrm{URT}(k)$, is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonstrate that $\mathrm{URT}(3)=\frac{7}{4}$. Then we show that $\mathrm{URT}(k)\ge \frac{k-1}{k-2}$ for all $k\ge 4$. We conjecture that $\mathrm{URT}(k)=\frac{k-1}{k-2}$ for all $k\ge 4$, and we confirm this conjecture for $k\in \{4,5,\dots ,21\}$. We then consider related problems in pattern avoidance; in particular, we find the undirected avoidability index of every binary pattern. This is an extended version of a paper presented at WORDS 2019, and it contains new and improved results.

对于有理数[R ，使得$\mathrm{1个}<\mathrm{[R}\le \mathrm{2个}$，无向r幂是形容词$XÿX^{\prime }$，其中单词x是非空的$X^{\prime }$ 在 $\{X，X^{\mathrm{[R}}\}$，而我们有 $|XÿX^{\prime }|/|Xÿ|=\mathrm{[R}$。该无向重复阈值用于ķ字母，表示$\mathrm{捷运}（ķ）$是所有r的集合的最小值，因此可以避免对k个字母产生无向r幂。我们首先证明$\mathrm{捷运}（3）=\frac{7}{4}$。然后我们证明$\mathrm{捷运}（ķ）\ge \frac{ķ-\mathrm{1个}}{ķ-\mathrm{2个}}$ 对全部 $ķ\ge 4$。我们推测$\mathrm{捷运}（ķ）=\frac{ķ-\mathrm{1个}}{ķ-\mathrm{2个}}$ 对全部 $ķ\ge 4$，我们确认了这个猜想 $ķ\in \{4，5，\dots ，\mathrm{21岁}\}$。然后，我们在模式规避中考虑相关的问题；特别是，我们找到每个二进制模式的无向可避免性指数。这是在WORDS 2019上发表的论文的扩展版本，包含新的和改进的结果。