A Novel Mathematical Exploration of Fractal Dynamics in Hyperbolic Spaces - Ansh Mishra; Sanjeev Kumar Pathak; Ashutosh Kushwaha (Recent Trends in Mathematics, 2026)
This research paper presents an original study on the behavior, generation, and properties of fractal structures within hyperbolic geometry. Unlike classical Euclidean fractals, hyperbolic fractals demonstrate accelerated boundary complexity and distinct scaling symmetries due to the curvature of the underlying space. The paper proposes new iterative models, analyzes geometric invariants, and explores potential applications in data visualization, network science, and theoretical physics. This research paper conducts an in-depth investigation into the formation, behavior, and mathematical characteristics of fractal structures embedded in hyperbolic geometry. Whereas traditional Euclidean fractals follow familiar patterns of self-similarity and scaling, fractals constructed in hyperbolic space exhibit fundamentally different behavior due to the negative curvature of the underlying geometry. These hyperbolic fractals grow with rapidly increasing boundary complexity, display unique scaling symmetries, and often reveal structural patterns that do not appear in flat space. To study these phenomena, the paper introduces a set of new iterative generation models designed specifically for hyperbolic environments. It further examines the geometric invariants that arise from these constructions and investigates how curvature influences fractal dimensionality, metric properties, and tiling-based growth dynamics. Beyond theoretical contributions, the work highlights several promising applications, including advanced data-visualization methods, improved representations for large-scale networks, and potential implications for models in theoretical physics that operate on curved or non-Euclidean spaces.
Keywords: Chaotic dynamics; fractal dynamics; geometric group theory; hyperbolic geometry; iterated function systems; nonlinear systems