We introduce and study in detail a special class of backward continued fractions that represents a generalization of Rényi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we construct the natural extension for the transformation that generates the Rényi-type expansion. We also define the random system with complete connections associated with the underlying dynamical system whose ergodic behaviour allows us to prove a variant of Gauss–Kuzmin-type theorem.

中文翻译：

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广义Rényi连分数的完全连接依赖法

我们详细介绍并研究了一类特殊的后向连分数，它代表了人一连分数的推广。我们研究了这些扩展中出现的数字的主要度量属性，并为生成 Rényi 类型扩展的转换构建了自然扩展。我们还定义了具有与底层动力系统相关联的完全连接的随机系统，其遍历行为使我们能够证明高斯-库兹明型定理的变体。