Fractals, digital expansions, words, and random structures

The project "Fractals, digital expansions, words and random structures" lies at the intersection of several fields of mathematics. The first part is devoted to fractals: the Sierpinski carpet and generalisations, and limiting net sets. The latter were introduced by the applicant, have the property of being ``well distributed`` in a certain sense and can be viewed as random fractals but also as generalised Sierpinski carpets. For limiting net sets we are interested, e.g., in aspects like topological and fractal geometric properties, geodesic distances between points, percolation, properties of their pore structure in the context of modelling porous materials with generalised Sierpinski carpets. We are also interested in generalisations of limiting net sets. The second part is dedicated to digital expansions, random words, and connections between them. Probability measures that are inductively defined on the unit interval and stem from the study of arithmetic functions, like the multinomial or the Gray code measure, induce probability measures on certain sets of random words, including some distributions corresponding to digital expansions with missing digits. One of our aims is to extend results already obtained (by the applicant or other authors) on these measures, and consider generalisations thereof. We intend to apply these types of constructions also in higher dimensions, and to construct and study measures whose support are fractals, e.g. some of those studied in the first part. Further problems regard combinatorics on words, Hamming weights, and discrepancies of point sets in the unit interval, also with respect to special measures.

The project's most significant results consist in creating new families of plane fractals, with special topological and geometrical properties. All these fractals are included in the unit square and are related to the Sierpinski carpet. We have three di?erent families of them: limit net sets, labyrinth fractals and generalised Sierpinski carpets. We used patterns in order to construct the fractals. In all these cases the patterns are, roughly speaking, obtained by dividing the unit square into n n squares with the same side length, and then deleting (colouring) some of these squares of side length 1=n and taking what remains.In the case of the limit net sets we use patterns where the deleted squares are evenly" distributed and provide an even" distribution of the holes in the fractals. We proved results regarding the use of special families of patterns in order to obtain fractals with di?erent degrees" of connectedness. In general, limit net sets are not self-similar and many of the results hold under pretty unrestricted conditions. The fractal dimension of these sets was also determined.The labyrinth fractals are self-similar dendrites that are constructed by using patterns whose graph has special properties (e.g., they are trees). We prove that under certain conditions on the patterns, for any two points in the labyrinth fractal, the path (in the fractal) that connects them has infinite length. We also prove other geometric properties of such paths.The generalised Sierpinski carpets are defined by using any patterns, infinitely many, with no restrictions (like those for the fractals mentioned above). We proved conditions on the patterns such that the fractal is connected. We also proved the existence of special families of patterns that provide, even in more general cases, totally disconnected generalised carpets.The above mentioned results and methods of work used here provide new information and tools for the case when such carpets are used as models in Theoretical Physics. Therefore I was invited to present these results and initiate a collaboration with theoretical physicists from abroad.Also, we were able to solve a problem that was open until now, about shortest paths in Sierpinski graphs and the distance between points in the Sierpinski gasket.Related to the other part of the project we obtained and published results on combinatorics on words, properties of measures on the unit interval related to digital expansions, and Gray codes (also related to fractals). Some of these are also related to problems from Theoretical Informatics.