Hausdorff dimension and Iterated Function Systems

In this thesis we introduce the concepts of Hausdorff dimensions and box-counting (or Minkowski-Bouligand) dimension, their properties and the similarities and differences between them. These dimensions can be seen as a generalization of usual integer-valued geometric dimensions (vector spaces, smooth manifolds, etc.), and they closely relate to the Hausdorff measure on Rn and how it changes. Ultimately, the main focus of this work is to prove a couple of theorems regarding specific fractal objects called Iterated Function Systems, and the calculation of their fractal dimension.