Supermixed labyrinth fractals

TL;DR

This paper introduces supermixed labyrinth fractals, a generalization of previous labyrinth fractals, and investigates their topological properties and arc lengths, extending earlier results to this broader class.

Contribution

It defines supermixed labyrinth fractals, studies their properties, and provides conditions for infinite arc length, expanding the theory beyond self-similar and mixed cases.

Findings

Extended formulas from previous cases to supermixed fractals.

Provided a sufficient condition for infinite arc length.

Analyzed topological and arc properties of supermixed labyrinth fractals.

Abstract

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.