AI 中文总结
针对满足四项特定条件的指数分离全纯迭代函数系，研究证明其关联的自共形测度维数及自共形集的豪斯多夫维数均达到自然上界，证明结合了分形测度维数理论与复分析方法。
AI 中文摘要
设Φ为复平面ℂ上有界区域内的全纯迭代函数系（holomorphic IFS），假设满足以下条件：（1）Φ中的映射无公共不动点；（2）不存在被Φ中所有映射保持的正则实解析曲线；（3）Φ不与相似迭代函数系（homothetic IFS）全纯共轭；（4）Φ是指数分离的。在这些假设下，本文证明与Φ相关的自共形测度的维数，以及对应的自共形集的豪斯多夫维数，均达到其自然上界。该证明结合了近期发展的平稳分形测度维数理论方法与复分析论证。
英文摘要
Let $Φ$ be a holomorphic IFS on a bounded domain in $\mathbb{C}$. Suppose that the following conditions hold: (1) the maps in $Φ$ do not have a common fixed point; (2) there does not exist a regular real-analytic curve which is invariant under all of the maps in $Φ$; (3) $Φ$ is not holomorphically conjugate to a homothetic IFS; (4) $Φ$ is exponentially separated. Under these assumptions, we show that the dimensions of the self-conformal measures associated to $Φ$, as well as the Hausdorff dimension of the associated self-conformal set, attain their natural upper bounds. The proof combines recently developed methods from the dimension theory of stationary fractal measures with complex-analytic arguments.
Comments41 pages
