Topology of fractale tiles

01.09.2010 - 30.09.2013
Research funding project
Mandelbrot introduced the word fractal in 1975 to describe shapes with a self-similar structure. Coast lines, snow flakes, clouds, crystals,... all these objects look similar at each level of magnification and are natural examples. Fractals also show up in chaos theory as attractors of dynamical systems. They have nowadays many applications, like data compression, computer graphics, diffusion processes, ... A fractal tile is a self-similar pattern with a tessellation property: replications of the tile can be fitted together to cover the space without overlap or gap, despite the fractal shape of the boundary. They appear in various branches of mathematics, such as number theory, dynamical systems or discrete geometry, and many problems in these areas can be restated in terms of topological properties of a fractal tile. Their study then requires methods from fractal geometry, complex analysis and automata theory. In this project, we propose to investigate large classes of planar fractal tiles: self-affine tiles, crystallographic replication tiles and substitution tiles. Self-affine tiles provide periodic tilings of the space by translations and they have been extensively studied. They can be seen as the fundamental domains of a numeration system, and their boundaries as the numbers with multiple expansions. Crystallographic reptiles also provide periodic tilings but allow the use of rotations. Substitution tiles were created by Rauzy in order to generalize the dynamics of interval exchange transformations to higher dimensions. They are often called Rauzy fractals and may provide periodic but also aperiodic tessellations. In all these classes, the induced tilings allow to derive topological properties of a tile from an analysis of its boundary. In our study, an important tool will be a boundary parametrization recently introduced by the applicant and a collaborator. The procedure was successful for many examples of self-affine tiles, including the canonical number system tiles. The parametrization sticks to the boundary of the tile in a measure theoretical sense. The essential point is that it can be followed by a finite state automaton. This will lead to deep topological information on the tile itself. Automata are in fact of common use in the theory of tilings generated by fractal tiles. However, they give rise to a symbolic description of the boundary of the tiles, and it is usually difficult to extract the topological information from the automata: this will be the rôle of the parametrization. The goals of the project are the following ones. We want to extend the parametrization procedure for the above mentioned classes of tiles without loosing the fundamental properties. The difficulties arise from the topological variety of the tiles among these classes. We wish to use the parametrization to explore the topological properties of tiles that are not disk-like. This will involve the implementation of algorithms in order to deal with the automata. Finally, we will explore the limits of the parametrization procedure and consider more general classes of fractal tiles. In the same time, we will produce new classes of crystallographic tiles by introducing crystallographic number systems. We think that this will generate tiles of a mild topological complexity, and therefore help us in answering the theoretical questions and in implementing the algorithms. All along this project, we will be able to test our progress on examples and subclasses whose topological properties are still unknown.

People

Project leader

Institute

Grant funds

  • FWF - Österr. Wissenschaftsfonds (National) Austrian Science Fund (FWF)

Research focus

  • Beyond TUW-research focus: 100%

Keywords

GermanEnglish
fraktalfractal
Pflastertile
Ziffernsystemnumber system
Automatenautomata

External partner

  • Montanuniversität Leoben

Publications