Homeomorphic spaces of two fractal models

Abstract

The article describes the iterative algorithms used to obtain fractal sets. It analyses and compares two algorithms: one implementing a classical approach, known as the chaos game, with constant parameters, and the other implementing an approach based on the construction of random partitions based on a set of elements of a convergent series. The results of the analysis allowed us to conclude that the set obtained using the second approach is an invariant of the constructed prefractal set. It is shown that using the second approach to the construction of fractal sets we can easily obtain homeomorphic prefractal sets. The topological properties of attractors obtained by constructing iterated random function systems are analysed. The article demonstrates that it is easier to determine some topological properties in the spaces obtained during the implementation of the second approach, and then map them on to the other spaces. The paper presents two models of biological communities characterised by the gregarious behaviour of animals. The behaviour of honeybees and pelagic fish during breeding and the formation of gregarious communities is considered as an example corresponding to the interpretations of the two different algorithms of fractal set construction considered in the article. It is demonstrated that the very nature of reproduction (genesis) and behaviour of individuals of these biological communities corresponds to different algorithms of the previously considered iterative random function systems. The differences in the behaviour of these gregarious communities are most noticeable in the behaviour of isolated individual parts of these communities.