The density of a rational language can be understood as the frequency of some pattern in the shift space, for example a pattern like "words with an even number of a given letter." We study the density of group languages, i.e. rational languages recognized by morphisms onto finite groups, inside shift spaces. We show that the density with respect to any given ergodic measure on a shift space exists for every group language, because it can be computed by using any ergodic lift of the given measure to a skew product between the shift space and the recognizing group. We then further study densities in shifts of finite type (with a suitable notion of irreducibility), and then in minimal shifts. In the latter case, we obtain a closed formula for the density under the condition that the aforementioned skew product has minimal closed invariant subsets which are ergodic under the product of the original measure and the uniform probability measure on the group. The formula is derived in part from a characterization of minimal closed invariant subsets for skew products between shifts and finite groups relying on notions of cocycles and coboundaries. In the case where the whole skew product is ergodic under the product measure, then the density is just the cardinality of the subset of the group which defines the language divided by the cardinality of the group. Moreover, we provide sufficient conditions for the skew product to have minimal closed invariant subsets that are ergodic under the product measure. Finally, we investigate the link between minimal closed invariant subsets, return words and bifix codes.

翻译：有理语言的密度可理解为移位空间中某种模式的频率，例如"给定字母出现偶数次的词"这类模式。我们研究群语言（即通过同态映射到有限群所识别的有理语言）在移位空间中的密度。证明表明：对移位空间上任意给定遍历测度，每个群语言的密度均存在，这是因为可通过将给定测度提升为移位空间与识别群之间的斜积上的任意遍历测度进行计算。进一步，我们研究了有限型移位（在适当的不可约性概念下）及极小移位中的密度。在后者情形下，我们得到密度的闭式公式，其成立条件为前述斜积具有在原始测度与群上均匀概率测度的乘积下遍历的极小闭不变子集。该公式部分源自对移位与有限群之间斜积的极小闭不变子集的刻画，该刻画依赖于上循环与上边界概念。当整个斜积在乘积测度下遍历时，密度等于定义该语言的群子集基数除以群基数。此外，我们给出了斜积具有在乘积测度下遍历的极小闭不变子集的充分条件。最后，我们研究了极小闭不变子集、返回词与双前缀码之间的联系。