Hausdorff and packing dimensions of Mandelbrot measure
Najmeddine Attia
Abstract
We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text]. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of a fractal set related to covering number on the Galton–Watson tree.
Topics & Concepts
MathematicsMandelbrot setHausdorff measureMeasure (data warehouse)Tree (set theory)Context (archaeology)CombinatoricsBoundary (topology)Hausdorff spaceFractalOuter measureHausdorff dimensionPacking dimensionFractal dimensionMinkowski–Bouligand dimensionMathematical analysisBiologyComputer scienceDatabasePaleontologyMathematical Dynamics and FractalsStochastic processes and statistical mechanicsTheoretical and Computational Physics