Fractal crystallographic tilings

Benoît Loridant

Vienna University of Technology & University Leoben, Austria

A compact set $ T$ tiles the plane with respect to a countable group of isometries $ \Gamma$ containing the identity map if

$\displaystyle \mathbb{R}^2=\bigcup_{\gamma\in\Gamma}\gamma(T),$

where the pieces $ \gamma(T)$ can only intersect at their boundaries. We are interested in sets $ T$ having also the property that an expanding affinity $ g$ blows up $ T$ onto a finite union of some of its isometric copies, i.e.,

$\displaystyle g(T)=\bigcup_{\displaystyle \delta\in\mathcal{D}}\delta(T)$

for some finite digit set $ \mathcal{D}\subset\Gamma.$ This endows the tile $ T$ with a self-similar structure. We wonder when it is homeomorphic to a closed disk. In fact, the topology of $ T$ is closely related to the configuration of the pieces it intersects in the tiling (its neighbors). The data $ (\Gamma,g,\mathcal{D})$ completely determines the neighbors: they can be computed algorithmically. Several easily checkable criteria of disk-likeness for $ T$ will be given, involving graphs in the general case, or the shape and number of the neighbors in particular cases.

Supported by the Austrian Science Foundation (FWF), projects Nr. S9604, S9610, and S9612.