Hausdorff dimension, its properties, and its surprises

Abstract

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension has some surprising properties: we construct a set $E\subset\C$ of positive planar measure and with dimension 2 such that each point in $E$ can be joined to $\infty$ by one or several curves in $\C$ such that all curves are disjoint from each other and from $E$, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in $\C$ which is not on one of these curves is in $E$. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as $z\mapsto π\sin(z)$.