Kifer, Peres, and Weiss proved in [A dimension gap for continued fractions with independent digits. Israel J. Math.124 (2001), 61–76] that there exists $c_{0}>0$, such that $\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$ for any probability measure $\unicode[STIX]{x1D707}$, which makes the digits of the continued fraction expansion independent and identically distributed random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.

中文翻译：

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最大化可数分支系统的伯努利度量和维度差距

Kifer、Peres 和 Weiss 在 [具有独立数字的连分数的维数差距中证明了这一点。以色列 J. 数学。124(2001), 61–76] 存在$c_{0}>0$, 这样$\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$对于任何概率测度$\unicode[STIX]{x1D707}$，这使得连分数展开式的数字独立且同分布的随机变量。在本文中，我们证明了在这类度量中，存在一个维度是最大的。我们的结果也适用于可数分支系统的更一般设置。