Topology of planar and higher dimensional self-replicating tiles

This is a joint research project with Prof. Shigeki AKIYAMA (Tsukuba, Japan) in the framework of a bilateral programm JSPS / FWF. Besides the intern collaboration Japan/Austria, further international collaborations will be involved to carry out the project (China, France, Japan, Korea, USA). Fractals are beautiful shapes that look similar at each level of magnification, like snow flakes, crystals, coast lines, Romanesco broccoli,... Since the introduction by Mandelbrot of a mathematical framework in 1975, fractals have found many applications in graphic arts, computer science, medicine, physics,... Fractal sets come up naturally in many fields of mathematics. For example, they describe the chaotic behaviour of polynomials in complex dynamics (Julia and Mandelbrot sets). They appear as fundamental domains of numeration systems (Knuth dragon). They also provide geometric representations of discrete dynamical systems (Rauzy fractals). Therefore, investigating the topological properties of the associated fractal sets allows to solve problems in various branches of mathematics such as discrete geometry, number theory or measure theory. However, as fractals are complex geometrical objects, their study is rather difficult and requires several tools from complex analysis, fractal geometry, automata theory. This project is devoted to the topological study of fractal sets leading naturally to covering of the d- dimensional space. Investigations of two-dimensional fractals with a wild topology can be found in the literature for isolated examples, but there is a lack of tools in order to make the investigation systematic and algorithmic. Also, we can find almost no result on three- or higher-dimensional fractals, as the complexity grows with the dimension. To attack these problems, we will develop 1) new methods and algorithms to characterize and classify higher-dimensional fractals; 2) a parametrisation method to get an access to very deep topological informations on planar fractals; 3) new classes of fractals, in particular associated with substitutions by weakening the usual fundamental assumptions on the substitutions or with discrete dynamical systems in relation with number theoretical problems.

This project was devoted to the development of new methods in the study of fractals. The objects we studied have a self-replicating structure: 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. We further investigated substitution tilings (the Penrose tiling is a famous example) and solved fundamental questions concerning the geometric realization of the dynamics and the relation between the dynamics and a notion of order in the tiling. The difficulty of studying these tiles arises from the fractal geometry of their boundaries. It requires methods from fractal geometry, discrete geometry, complex analysis, tiling theory and automata theory. Investigations of two-dimensional fractals with a wild topology can be found in the literature for isolated examples, but there was a lack of tools in order to make the investigation systematic and algorithmic. In the project, we could solve several classification problems of fractal families. We also set up a new method to study Pisot substitutions that fail to be irreducible. It involves higher-dimensional extensions of substitutions, for which we found geometric representations in terms of stepped surfaces, self-replicating and periodic tilings made of Rauzy fractals. We further studied fundamental properties of tiling spaces and connected a notion of order of the associated Delone sets with its dynamical properties. Finally, we investigated the topology of general planar compact sets and proved fundamental results on their core decomposition. The project was embedded in a JSPS/FWF joint project. A strong cooperation with the Japanese partner Shigeki AKIYAMA (Tsukuba) as well national (within the University of Leoben) and further international collaborations (China, France, Korea) made the progress successful, by combining the topological background of the Austrian side with the dynamical expertise of the Japanese side. Two project meetings at St Virgil by Salzburg and a symposium on related topics during the ÖMG/DMV congress in 2017 were organized. We were also granted an ÖAD/ANR fund to strengthen the collaboration with France in related research topics.