We study 4 × 4-labyrinth fractals, which are self similar dendrites. For all 4 × 4-labyrinth fractals we answer the question, whether there is a curve of finite length in the fractal from one point to another point in the fractal. In the first case, between any two points in the fractal there is a unique arc a, the length of a is infinite, and the set of points, where no tangent exists to a, is dense in a. In the second case, there are also pairs of points between that there is a unique arc of finite length.

Labyrinth fractals are self-similar fractals that were introduced and studied in recent work [2, 3]. In the present paper we define and study more general objects, called mixed labyrinth fractals, that are in general not self-similar and are constructed by using sequences of labyrinth patterns. We show that mixed labyrinth fractals are dendrites and study properties of the paths in the graphs associated to prefractals, and of arcs in the fractal, e.g., the path length and the box counting dimension and length of arcs. We also consider more general objects related to mixed labyrinth fractals, formulate two conjectures about arc lengths, and establish connections to recent results on generalised Sierpiński carpets.

Generalised Sierpiński carpets are planar sets that generalise the well-known Sierpiński carpet and are defined by means of sequences of patterns. We study the structure of the sets at the kth iteration in the construction of the generalised carpet, for k ≥ 1. Subsequently, we show that certain families of patterns provide total disconnectedness of the resulting generalised carpets. Moreover, analogous results hold even in a more general setting. Finally, we apply the obtained results in order to give an example of a totally disconnected generalised carpet with box-counting dimension 2.

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.

In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t, m, s)-nets over Z 2 which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the root mean square worst-case error converges with order 2 m(−2+ε) for any ε > 0, where 2 m is the number of points. A similar result for lattice rules has previously been shown by Hickernell.