Fractals and Numeration

In Austria as well as in France research on fractal sets with special emphasis on fractals coming from various kinds of numeration has a long tradition. The aim of this project is to bundle the forces existing in both countries and to exploit the synergies emerging from putting together the different viewpoints on fractals and numeration. This enables us to consider the topic of this proposal in a much broader way than it would be possible for a project carried out on a national level. We consider these synergies as a strong feature and a definite added value of the present project. The present project aims at studying fractal sets arising from various numeration systems. To this matter, we subdivide our project into four tasks, each of them throwing a different light on our topic. Arithmetics, dynamics and expansions (ADE) Topological properties of fractals (ToF) Rauzy fractals and substitutions (RaSub) Fractals with a view towards applications (FracApp) In the first task (ADE), we are concerned with arithmetic properties of numeration. The topics here vary from transcendence properties in the spirit of the van der Poorten-Loxton Theorem which has been proved recently by two members of the French team [AB3], over natural extensions of continued fraction algorithms, to numeration in algebraic number fields, to redundant number systems and their applications in cryptography. In these problems about generalized numeration is intimately linked to geometric and topological properties of underlying fractals. Thus in the second task (ToF), we will investigate these properties. Although there is a vast literature on topological properties of fractal sets, most results are valid only for the twodimensional case. In close contact with the research group of the topologists Jim Cannon and Greg Conner from Brigham Young University (Utah, USA) we wish to break this limitation and obtain new results on the topology for self-affine fractals in higher dimensions. Moreover, we wish to exploit the relevance of these results to the arithmetic of the number systems studied in ADE. While the fractal sets in first two tasks are related to numeration from a very general point of view, the third task (RaSub) is devoted to the special case of Rauzy fractals and their associated substitutions. Here, using the general results of the previous tasks, we wish to gain new results on Rauzy fractals especially in the more general context of S-adic expansions. For instance, these fractals will give us new insight on generalized continued fraction algorithms. It is therefore desirable to explore as many of their properties as possible. Of course, we also wish to relate our research to the Pisot conjecture and illustrate the conjecture through a variety of new methods and techniques coming together in this project. In the last task (FracApp) we will consider various properties of fractals like the intersection of fractals with lines and fractal homeomorphisms that play a role in image processing. This part of the project can be regarded as an interface to applications of fractals in other branches of science. It shows that the ideas and methods developed in this project are interesting also in other branches of science. Besides that, we also wish to explore an interesting relation between fractal transformations in image processing and number theory.

Arithmetics, numeration, and fractals For hundreds of years, the decimal system was used to encode natural and real numbers. In the last decades, the binary system, as well as more exotic numeration systems, proved to be indispensable tools in mathematics and in computer science. Let us quote for instance arithmetic properties of q-ary expansions (with a rational integer q ? 2) in elementary number theory, or else expansions in Gaussian integers, considered by Knuth in the special situation q = -1+i. While the fundamental domain associated with a rational integer base is always equal to an interval, for this complex base we obtain the well-known twin dragon fractal as a fundamental domain. This illustrates the link between radix representations and fractal sets. More generally, the self-similar structure inherent to radix representations reflects to self-similar geometrical objects that are naturally associated with these numeration systems: number systems are fractals in the large. The present project is devoted to the study of fractal sets issued from numeration systems by focusing on their arithmetic properties.Dynamical systems and numeration In Austria as well as in France research on fractal sets with special emphasis on fractals coming from numeration systems has a long tradition, involving various kinds of problems: transcendence properties, continued fraction algorithms, numeration in algebraic fields, beta-numeration, etc. These problems all have links with the topological and geometric properties of fractal sets, and in particular with the topology of self-affine fractals. This project links important themes of discrete mathematics, dynamics and number theory, including geometric and topological aspects, as well as computer applications. We have focused on four topics, namely links between arithmetics, dynamical systems and expansions, topological properties of fractals, links between Rauzy fractals and substitutions, and finally, fractals with a view towards applications. The dynamical system approach developed here is a key element in the study of fractal structures. Exploiting the synergies emerging from putting together the different viewpoints on fractals and numeration in the two countries allowed us to consider the subject in a much broader way.Among the various achievements of this project, our main realizations are in the context of the study of the Pisot dynamics. Rauzy fractals are naturally associated with Pisot-type substitutive dynamical systems. We have studied them in the S-adic framework, extending the substitutive framework to parameters that are not necessarily algebraic. This has far reaching consequences for continued fraction algorithms. We also obtained topological properties concerning the study of the fundamental groups of Rauzy fractals.