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Conditions for homeomorphism of sets modeled by randomized iterated function systems

Numerical modeling of fractal sets using randomized iterated function systems can be implemented in two ways. The first of them is known as the chaos game, and the other is associated with the construction of a transformation matrix, the elements of which are the sums of terms of a converging series, divided by given classes. The last way allows us to relatively easily generate realizations of homeomorphic prefractal sets with similar topological properties.