Fractals and Words: Topological, Dynamical, and Combinatorial Aspects

The present project lies at the intersection of several fields of mathematics: Fractal Geometry, Combinatorics, Topology, Number Theory, Graph Theory, Measure Theory Dynamical Systems, and others. The main part of the project is devoted to fractals. First, we deal with families of Sierpinski carpets, several of which were defined and studied in recent publications of the applicant. We also introduce several new objects related to these. Connectedness properties, geodesic paths and distances between points in the fractals, percolation, addresses, and graphs associated to the (pre)fractals are some of the topics we want to study. We also want to see how the patterns that generate these fractals influence several of these properties. We are interested in extensions of the results to larger classes of fractals, also in higher dimensions and also in new aspects, like using the frame of V-variable fractals or random fractals. We also want to use these carpets or adapt them for modelling porous materials and diffusion. Self-affine tiles were introduced by Thurston and Kenyon more than 20 years ago and have been extensively studied since then. Concerning their topology, mainly planar tiles have been understood so far. A recent paper by Conner and Thuswaldner brings a breakthrough here and allows to study their topology in higher dimension. We will push forward this research and study topological properties of self-affine tiles and Rauzy fractals in higher dimension. S-adic words and their dynamics are linked by Rauzy fractals that go back to the seminal work of Rauzy in the beginning 1980s. We wish to use new progress on the geometric theory of S-adic words and Rauzy fractals to construct nonstationary Markov partitions in the sense of Arnoux and Fisher and to get natural codings of rotations on higher dimensional tori that have linear complexity. The second part is dedicated to combinatorics on words, digital expansions and codes, in particular Gray codes. Probability measures that stem from the study of arithmetic functions and were inductively introduced in the unit interval based on digital expansions, are here generalised and also defined on some fractals. There also problems of discrete dynamical systems occurring in this context. Moreover, several connections between the section on digits and words and that dedicated to fractals occur in the project proposal.. The results of this research have implications not only in Pure Mathematics, but also in modelling in Theoretical Physics, in Theoretical Informatics and Simulation.

In this project, new fractals were introduced and studied with respect to their topological and geometric properties. Therefore new tools and methods were introduced, and combined with already existing ones, from several areas of mathematics. We created mixed and supermixed labyrinth fractals, constructed iteratively based on sequences of patterns or sequences of collections of patterns, respectively. These are square patterns, obtained by dividing the unit square into smaller congruent squares, coloured with two colours. We get generalised Sierpinski carpets that are fractal dendrites, and are in general not self-similar or self-affine. Self-similar fractal dendrites occur in other authors' recent research in the context of Iterated Function Systems. Mixed and supermixed labyrinth fractals, equipped with the tools and methods developed in the project, bridge a gap between self-similar dendrites or carpets and randomised ones. We establish criteria on how the patterns' shape has implications on the length or dimension of arcs in the fractals, or on their iterative construction. Our results, e.g., on paths in pre-fractals offer valuable tools for a computational approach, and lead to a new international collaboration, with a theoretical physicist, on features of mixed labyrinth fractals as models in theoretical physics. Moreover, our results and ideas on labyrinth fractals recently lead to the construction, by other theoretical physicists abroad, of prototypes of radar antennas. We also introduced self-similar triangular labyrinth fractals. We obtained new features, based on the triangular shape and certain duality, and could establish criteria for having finite, infinite, or both finite and infinite arcs in the fractals, and the fractal dimension of arcs in these dendrites. Generalisations thereof are in preparation for submission. New topological results on three-dimensional self-similar tiles were obtained. This is inasmuch remarkable as most of the previous results in this direction hold only in the two-dimensional space. In the three-dimensional space there is no analogon of the Jordan Curve Theorem available. As a consequence, new ideas are necessary in order to prove topological results. With the help of a theory of R.L. Bind from the 1960s combined with modern methods from fractal geometry results about self-affine tiles with spheric boundary and, respectively, about self-affine tiles that are homeomorphic to a sphere were gained. Moreover, S-adic words were studied, which are related to multidimensional continued fractions and Rauzy fractals, and have interesting combinatorial properties. These words were used to code Kronecker rotations on the torus. We made all published results free available online to all interested scientists. The project made many national and international collaborations possible that lead to important progress in the research. Moreover, the project's results already inspired a new, rich research agenda for the future, to further broaden the theoretical framework developed in this project.