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We have found that in a system consisting of small magnetic particles a phenomenon related to the formation of fractal structures may arise. The fractal features may arise not only in the distribution of magnetic moments but also in their energy spectrum. The magnetization and the susceptibility of the system also display fractal characteristics. The multiple structures are associated with exponentially many locally stable minima in a highly complex energy landscape. The signature of these fractal structures can be experimentally detected by various methods.