A Remarkable Insight in the Theory of Fractals Confirmed Through Counterexamples - Ri, Song-Il; Drakopoulos, Vasileios; So, Yong-Nam; Kim, Gol (Discontinuity, Nonlinearity, and Complexity, 2026)
Based on the mutual relations among the well-known Banach contraction mapping (1922), Rakotch contraction mapping (1962), Kannan contraction mapping (1968), Bryant contraction mapping (1968), and Reich contraction mapping (1971) - particularly considering corresponding examples, purposeful and deliberate counterexamples, and the remarkable analytic properties of the Bryant contraction mapping, which is the most explicit and simple generalisation of the Banach contraction mapping - we confirm, for the first time, that only a few contractive mappings, such as the Rakotch contraction mapping, can generate fractals. At the same time, we establish that one cannot always generate fractals using various generalised Banach contraction mappings in a standard Euclidean metric space, as most of the analytic properties of these contraction mappings on the base space are not necessarily inherited by the mappings they generate on the fractal space.