Hausdorff dimension, its properties, and its surprises
| dc.creator | Schleicher, Dierk | |
| dc.date | 2005-05-05 | |
| dc.date | 2007-08-21 | |
| dc.date.accessioned | 2026-07-07T08:24:29Z | |
| dc.date.available | 2026-07-07T08:24:29Z | |
| dc.description | 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 $z\mapsto π\sin(z)$. | |
| dc.description | 28 pages, 4 figures. Minor revision in process of publication | |
| dc.identifier | https://arxiv.org/abs/math/0505099 | |
| dc.identifier | http://arxiv.org/abs/math/0505099 | |
| dc.identifier | American Mathematical Monthly 114 (2007), 509-528 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/136358 | |
| dc.subject | Dynamical Systems | |
| dc.subject | 37-02, 37F10, 37F35, 30D05, 54G20, 28A78 | |
| dc.title | Hausdorff dimension, its properties, and its surprises | |
| dc.type | text |