Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length (Cristea and Steinsky in Geom Dedicata 141(1):1–17, 2009; Proc Edinb Math Soc 54(2):329–344, 2011). In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.

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关于迷宫分形中的弧长

迷宫分形是单位正方形中的自相似树突，在迷宫集或迷宫图案的帮助下定义。在分形由水平和垂直阻塞模式生成的情况下，分形中任意两点之间的弧具有无限长（Cristea 和 Steinsky 在 Geom Dedicata 141(1):1–17, 2009; Proc Edinb Math Soc 54(2):329–344, 2011)。在混合迷宫分形的情况下，使用一系列迷宫图案来构建枝晶。在本文中，我们关注混合迷宫分形点之间的弧长。我们表明，根据序列中模式的选择，两种情况都可能发生：分形任意两点之间的弧的长度是有限的，或者分形任意两点之间的弧的长度是无限的。