An exploration of fractal dimension

Abstract

When studying geometrical objects less regular than ordinary ones, fractal analysis becomes a valuable tool. Over the last 30 years, this small branch of mathematics has developed extensively. Fractals can be de fined as those sets which have non-integral Hausdor ff dimension. In this thesis, we take a look at some basic measure theory needed to introduce certain de finitions of fractal dimensions, which can be used to measure a set's fractal degree. We introduce Minkowski dimension and Hausdor ff dimension as well as explore some examples where they coincide. Then we look at the dimension of a measure and some very useful applications. We conclude with a well known result of Bedford and McMullen about the Hausdor ff dimension of self-a ffine sets.