Several Methods for Calculating and Estimating the Hausdorff Dimension

Abstract

Among various fractal dimensions, the Hausdorff dimension is the most widely used and fundamental one. However, in many cases, calculating or estimating its value can be challenging. This article systematically introduces several methods for calculating the Hausdorff dimension, starting with its definition and properties. In method one, the dimension is estimated using upper and lower bounds. For the upper bound, it usually requires finding a specific covering, while for the lower bound, the mass distribution principle is commonly employed for estimation; In method two, the clever use of potential theory is applied to estimate the Hausdorff dimension; In method three, for some high-dimensional cases, the dimension can be reduced using the projection theorem, and then the original dimension is estimated by computing the dimension of the lower-dimensional cases; In method four, the dimension estimation of the Cartesian product of two sets is considered, and several theorems are used to provide upper and lower bounds for the dimension; In method five, the focus is on the dimension calculation of self-similar sets. For such sets, under the condition of open sets, heuristic methods can be used for calculation; In method six, analogous to self-similar sets, the estimation of the dimension for self-affine sets is summarized.