Fractals on the hyperbolic plane using Chebyshev style scaling functions

Abstract. In this paper, we elaborate on fractals within the context of the hyperbolic plane, using the complex unit disk as a model—specifically, the Poincaré disk representation of
Bolyai—Lobachevsky hyperbolic geometry. Blaschke functions act as congruence transformations in this setting. We introduce a novel family of scaling operations based on the area hyperbolic tangent function, constructed in a manner analogous to the explicit formulation of Chebyshev polynomials. We investigate the mathematical properties of these operations and demonstrate their application in generating fractals viewed as fixed points of Hutchinson maps. The resulting visualizations are produced using custom MATLAB programs. Throughout the paper, we also reflect on related contributions and insights by Professor Ferenc Schipp.