Introducción a la dimensión de Hausdorff

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: María Jesús Carro RossellIn the early-to-mid 20th century, several notions of dimension appeared to deal with some mathematical objects considered until that moment as “pathological”and not worthy of study. The aim of this project is to explore the notions of Hausdorff, Minkowski and box-counting dimension and their basic features. We show that Minkowski and box-counting dimensions are equivalent and that the Hausdorff dimension of a set does not exceed its Minkowski dimension. We also introduce self-similar sets, where the theory mentioned before becomes elegant and simple and where both dimensions match. Finally we construct a space-filling curve, namely, the Hilbert’s curve