In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that their limit sets are porous in large (in the sense of category and dimension) subsets. We also provide natural geometric and dynamic conditions under which the limit set of a GDMS is upper porous or mean porous. On the other hand, we prove that if the limit set of a GDMS is not porous, then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction.We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterisation of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are uniformly upper porous and mean porous almost everywhere.We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

中文翻译：

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共形动力系统中的孔隙度

在本文中，我们研究了保形分形孔隙度的各个方面。我们首先在无限图定向马尔可夫系统 (GDMS) 的一般背景下探索孔隙度，并且我们表明它们的极限集在大（在类别和维度的意义上）子集中是多孔的。我们还提供了自然几何和动态条件，其中 GDMS 的极限集是上多孔或平均多孔。另一方面，我们证明如果 GDMS 的极限集不是多孔的，那么它几乎在所有地方都不是多孔的。我们还重新审视了有限图有向马尔可夫系统的孔隙度，并提供了可检查的标准，以保证极限集在规定方向的每个尺度上都有相对大小的孔。然后，我们将关注范围缩小到与具有任意字母的复杂连分数相关的系统，并为它们的极限集提供了一种新的孔隙度表征。此外，我们介绍了高斯整数子集的上密度和上盒维数的概念，并探讨了它们与孔隙度的联系。作为应用，我们展示了复连分数系统的极限集，其字母表是余有限的，甚至是高斯素数的余有限子集，几乎在所有地方都不是多孔的，而它们几乎在所有地方都是均匀的上多孔和均值多孔。我们最后将注意力转向复杂的动力学，我们深入研究 Julia 亚纯函数集的孔隙度。我们表明，如果驯亚纯函数的 Julia 集不是整个复平面，那么它在其点的密集集上是多孔的，并且相对于自然遍历度量，它几乎处处都是多孔的。另一方面，如果 Julia 集不是多孔的，那么它几乎在所有地方都不是多孔的。特别是，如果函数是椭圆的，我们将证明它的 Julia 集在其点的密集集上不是多孔的。