On the length of arcs in labyrinth fractals

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.