线性与非线性无限IFS的极限集与复连续分数相关
The Limit Sets of Linear and Nonlinear Infinite IFSs Related to Complex Continued Fractions
AI总结：
本文研究了与复连续分数相关的线性与非线性无限IFS的极限集测度性质，揭示了其在Hausdorff和packing维度上的独特现象及维度差异。
摘要：
我们引入了两个由正实数序列d和自然数T参数化的无限迭代函数系统（IFS）族F(d,T)和G(d,T)，并研究了它们极限集的测度理论性质。G(d,T)是一个无限莫比乌斯IFS，是将实数连续分数的IFS扩展到复平面上闭单位圆盘上的IFS。F(d,T)是一个无限线性IFS，其第一个近似与G(d,T)相同。我们证明对于许多d和T的选择，F(d,T)和G(d,T)的极限集表现出仅存在于无限IFS中的现象，如在Hausdorff维度处具有零Hausdorff测度，在packing维度处具有无限packing测度，或Hausdorff维度和packing维度不同。我们还证明在某些参数条件下，F(d,T)的极限集的Hausdorff维度严格大于G(d,T)的。
英文摘要：
We introduce two families of infinite iterated function systems (IFSs) $\mathcal{F}(\mathbf{d}, T)$ and $\mathcal{G}(\mathbf{d}, T)$, parametrized by a sequence of positive real numbers $\mathbf{d}$ and a natural number $T$, and investigate the measure-theoretic properties of their limit sets. $\mathcal{G}(\mathbf{d}, T)$ is an infinite Möbius IFS, which is an extension of the IFS of real continued fractions to the IFS on the closed unit disc in the complex plane. $\mathcal{F}(\mathbf{d}, T)$ is an infinite linear IFS that shares the same first approximation as $\mathcal{G}(\mathbf{d}, T)$. We show that for many choices of $\mathbf{d}$ and $T$, the limit sets of both $\mathcal{F}(\mathbf{d}, T)$ and $\mathcal{G}(\mathbf{d}, T)$ exhibit phenomena unique to infinite IFSs, such as having zero Hausdorff measure at the Hausdorff dimension, having infinite packing measure at the packing dimension, or having different Hausdorff and packing dimensions. We also prove that the Hausdorff dimension of the limit set of $\mathcal{F}(\mathbf{d}, T)$ is strictly larger than that of $\mathcal{G}(\mathbf{d}, T)$ under certain conditions on the parameters.
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