超混合迷宫分形
Supermixed labyrinth fractals
摘要：
拉巴斯分形是单位正方形中的枝晶。在过去的十年中，它们首先在自相似情况下被引入和研究[Cristea & Steinsky（2009，2011）]，然后在混合情况下[Cristea & Steinsky（2017），Cristea & Leobacher（2017）]。超混合分形构成了混合迷宫分形的一个重要概括：迭代构造的每一步都不是根据一种迷宫模式完成的，而是可能根据几种不同的模式完成的。在本文中，我们介绍和研究supermixed迷宫分形和相应的prefractals，称为supermixed迷宫集，重点放在以前研究的自相似和混合的情况下：拓扑性质和性质的弧点之间的分形。这里的事实和公式推广了在上述情况下证明的结果。主要结果之一是得到了混合迷宫分形中弧长为无穷长的一个充分条件。
英文摘要：
Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [Cristea & Steinsky (2009,2011)], then in the mixed case [Cristea & Steinsky (2017), Cristea & Leobacher (2017)]. Supermixed fractals constitute a significant generalisation of mixed labyrinth fractals: each step of the iterative construction is done according to not just one labyrinth pattern, but possibly to several different patterns. In this paper we introduce and study supermixed labyrinth fractals and the corresponding prefractals, called supermixed labyrinth sets, with focus on the aspects that were previously studied for the self-similar and mixed case: topological properties and properties of the arcs between points in the fractal. The facts and formulae found here extend results proven in the above mentioned cases. One of the main results is a sufficient condition for infinite length of arcs in mixed labyrinth fractals.
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