For any $ \delta >0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=\delta$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.

中文翻译：

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Speiser 类 Julia 集的维数接近于 1

对于任何 $\delta >0$，我们构造一个完整的函数 $f$，其中包含三个奇异值，其 Julia 集最多具有 Hausdorff 维数 $1=\delta$。Stallard 证明，只要 $f$ 有一个有界奇异集，维度就必须严格大于 1，但是之前没有已知有限奇异集和维度严格小于 2 的例子。