Topology of fractal tiles

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. We apply for one full and one half post-doctorate positions and for a period of 36 months. The project will be carried out at the Department of Mathematics und Informationstechnologie of the Montanuniversity of Leoben and at the Institute for Discrete Mathematics and Geometry of the Technical University of Vienna. The full position is intended for the applicant who already worked as research assistant at those departments (2006-2008). The research will also involve a international collaboration network. Our results will be published in international journals and presented in international workshops and conferences.

This project was devoted to the development of new methods in the study of fractal tiles. Fractals are objects with self-affine structure, i.e., they contain smaller copies of themselves: snowflakes, crystals, coastlines, Romanesco broccoli, clouds, are examples of fractals that are encountered in the nature. They also show up in many fields of mathematics: in chaos theory as attractors of dynamical systems, in number theory, in discrete geometry,. They have many applications, in computer graphics, diffusion processes, image compression,. Thanks to this project, large classes of fractals could be classified by means of their topological properties.The difficulty of studying these tiles arises from the fractal geometry of their boundaries. In this project, a method called boundary parametrization was developed. It requires methods from fractal geometry, complex analysis, tiling theory and automata theory. With the help of the parametrization, it is possible to decide whether planar fractals can be continuously deformed to a disk or not. The analysis of non disk-like fractals is also possible via the parametrization, as it sticks to the fractal geometry of the boundary. Automata are commonly used when considering fractal tiles. However, these automata lead only to a symbolic description of the boundary, and it is difficult to extract the topological information from them. The boundary parametrization plays this role.Other goals could be reached in this project. New classes of fractals were created, leading to crystallographic tilings of the space. Deep topological information was obtained for fractal tiles that are not disk-like, for example, information on the so-called cut sets or on the components constituting the tiles with a wild topology. Finally, general classes of topological sets were considered and we introduced a numerical scale in order to quantify in which extent a set is far from the property of being locally connected.The project involved both national (TU Wien/MU Leoben) and international collaborations (France, China, Japan). In particular, from 2013 on, a tight collaboration was initiated with France through the FWF/ANR project Fractals and numeration (2013-2017). The project leader is a current member of this project.