Degenerate Fractals: A Formal and Computational Framework for Zero-Dimension Attractors

Ion Andronache ()
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Ion Andronache: Advanced Digital Archaeological-Historical Network, Alma Mater Europaea (AMEU)–ECM, Slovenska Ulica No. 17, 2000 Maribor, Slovenia

Mathematics, 2025, vol. 13, issue 21, 1-25

Abstract: This paper analyzes the extreme limit of iterated function systems (IFSs) when the number of contractions drops to one and the resulting attractors reduce to a single point. While classical fractals have a strictly positive fractal dimension, the degenerate case D = 0 has been little explored. Starting from the question “what happens to a fractal when its complexity collapses completely?”, Moran’s similarity equation becomes tautological ( r s = 1 with solution s = dim M = 0 ) and that only the Hausdorff and box-counting definitions allow an exact calculation. Based on Banach’s fixed point theorem and these definitions, we prove that the attractor of a degenerate IFS is a singleton with dim H = dim B = 0 . We develop a reproducible computational methodology to visualize the collapse in dimensions 1–3 (the Iterated Line Contraction—1D/Iterated Square Contraction—2D/Iterated Cube Contraction—3D families), including deterministic and stochastic variants, and we provide a Python script 3.9. The theoretical and numerical results show that the covering box-counting retains unity across all generations, confirming the zero-dimension element and the stability of the phenomenon under moderate perturbations. We conclude that degenerate fractals are an indispensable benchmark for validating fractal dimension estimators and for studying transitions to attractors with positive dimensions.

Keywords: fractal dimension; degenerate fractals; IFS; limit case; unique attractor; singleton attractor (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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