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Discontinuity, Nonlinearity, and Complexity 15(2) (2026) 157--167 | DOI:10.5890/DNC.2026.06.002

Dhilshana, Sunil Mathew

Department of Mathematics, National Institute of Technology Calicut, India-673601

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Abstract

This study investigates the properties, interactions, and attractors of irregular iterated function systems (IFS). Unlike traditional IFS, which rely on uniform contraction functions and symmetrical attractors, irregular IFS capture complex structures characterized by non-uniform scaling and unpredictable patterns. We show that the Hausdorff distance between the attractors of the product of irregular IFS can be bounded by the maximum contraction factors of the coordinate irregular IFSs. This extends classical IFS results to irregular settings. The Hutchinson operator for the product of irregular IFS uniquely fixes a point, corresponding to the product of the coordinate attractors, preserving the structure of attractors even in irregular cases. A modified Collage Theorem for product irregular IFSs estimates the Hausdorff distance between an arbitrary set and the attractor, providing tools for approximating fractal sets in higher dimensions. These findings provide new insights into the behavior of irregular IFS and their products, enhancing the understanding of composite fractal structures.

Acknowledgments

\bibitem{HFS} Hutchinson, J.E. (1981), Fractals and self similarity, \textit{Indiana University Mathematics Journal}, \textbf{30}, 713-747.

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