Hausdorff dimension, its properties, and its surprises

TL;DR

This paper reviews the properties of Hausdorff dimension, illustrating both expected and surprising phenomena through examples, including constructions related to complex dynamics and iteration of simple maps.

Contribution

It introduces novel examples of sets with counterintuitive Hausdorff dimension properties, linking them to complex dynamics and recent discoveries.

Findings

Existence of sets with positive measure and dimension 2 with surprising curve structures

Construction of sets where points are connected by disjoint curves with union dimension 1

Relevance of these examples to complex dynamics and iterative maps

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 $z\mapsto \pi\sin(z)$.