Fractals and Numeration – FAN

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 two-dimensional 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.