FRACTAL DIMENSION ESTIMATION AND ATTRACTOR EXISTENCE IN MR-METRIC SPACES: A GENERALIZATION OF CLASSICAL FRACTAL GEOMETRY

Abstract

This paper establishes new theoretical foundations for fractal analysis in MR-metric spaces, a ternary generalization of classical metric spaces. We prove two fundamental theorems: the first provides precise bounds for the Hausdorff dimension of fractal sets in MR-metric spaces, while the second guarantees the existence and uniqueness of fractal attractorsunder generalized contraction mappings. The results extend classical fractal geometry tothis broader setting and are demonstrated through explicit constructions of MR-modiffed Cantor sets and Sierpinski fractals, revealing how the ternary metric structure inuencestheir dimensional properties and self-similar characteristics.