Abstract
Fractal geometries are complex, and one way to assess them is to assign a suitable measure. The Hausdorff measure of a fractal gives an idea of the complexity of the geometry. Further, the measure is widely associated with the geometry of the fractal. One of the general approaches to calculate the Hausdorff dimension is by defining the optimal Hausdorff measure through a box-counting approach. The paper evaluates the Hausdorff dimension of the Mandelbrot set at various resolutions of the geometry. The convergence of the box-counting algorithm is studied. The paper also delves into the evolution of the geometry of the dynamical system zₙ₊₁ = zₙᵐ + c for the powers other than m = 2. The paper further discusses the convergence behavior of the box-counting algorithm for higher powers of m.
Keywords: Fractals, Hausdorff Dimension, Mandelbrot Set, Self-Similar Sets, Topology
How to cite
Deepak Agarwal (2025). Evaluation of Hausdorff Measure using Numerical Methods at Various Points and Resolutions of Mandelbrot Sets. International Journal of Mathematics and Computational Frontiers, 1(2), 59–70.