Recall that a Borel measure $mu$ on $mathbb{R}$ is said to be extremal if $mu$-almost every number in $mathbb{R}$ is not very well approximable. In this paper, we investigate extremality (implied by the exponentially fast decay (efd) property) of conformal measures induced by regular infinite conformal iterated function systems. We then give particular attention to the class of such systems generated by the continued fractions algorithm with restricted entries. It is proved that if the index set of entries has bounded gaps, then the corresponding conformal measure satisfies the efd property and is extremal. Also a class of examples of index sets with unbounded gaps is provided for which the corresponding conformal measure also satisfies the efd property and is extremal.
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机译：回想一下，如果$ mu $-$ mathbb {R} $中的几乎每个数字不是很好地近似的，则对$ mathbb {R} $进行Borel度量$ mu $被认为是极值。在本文中，我们研究了由规则无限保形保角迭代函数系统诱导的保形量度的极值（由指数快速衰减（efd）性质表示）。然后，我们特别注意由具有受限条目的连续分数算法生成的此类系统的类别。事实证明，如果条目的索引集具有有限的间隙，则相应的共形测度满足efd属性并且极值。还提供了一类具有无限制间隙的索引集的示例，对于该索引集，相应的保形度量也满足efd属性并且极好。
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