We study aperiodic balanced sequences over finite alphabets. A sequence v of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y'. We show that the language of v is eventually dendric and we focus on return words to its factors. We deduce a method computing critical exponent and asymptotic critical exponent of balanced sequences provided the associated Sturmian sequence u has a quadratic slope. The method is based on looking for the shortest return words to bispecial factors in v. We illustrate our method on several examples, in particular we confirm a conjecture of Rampersad, Shallit and Vandomme that two specific sequences have the least critical exponent among all balanced sequences over 9- resp. 10-letter alphabets.

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关于平衡序列及其临界指数

我们研究了有限字母表上的非周期性平衡序列。这种类型的序列 v 完全由 Sturmian 序列 u 和两个恒定间隙序列 y 和 y' 表征。我们表明 v 的语言最终是树状的，我们专注于将单词返回到它的因子。我们推导出一种计算平衡序列的临界指数和渐近临界指数的方法，前提是相关的 Sturmian 序列 u 具有二次斜率。该方法基于寻找 v 中双特殊因子的最短返回词。 我们通过几个例子来说明我们的方法，特别是我们证实了 Rampersad、Shallit 和 Vandomme 的猜想，即两个特定序列在所有平衡序列中具有最小的临界指数超过 9-resp。10 个字母的字母。