### Project 08: "Number systems and fractal structures"

*This project is running since 2010.*

Principal investigator: Jörg Thuswaldner

University of Leoben, Austria.

*Mentor for*: Ferizovic, Scheerer; Greinecker, Krenn, Weitzer.

Associated scientist: Peter Kirschenhofer (since 2015)

University of Leoben, Austria.

*Mentor for*: Zhang; Minervino, Schmuck.

### DK Students

**Second phase of the doctoral program:****Shu-Qin Zhang**(China; since October 2015)

*Email*: shuqin.zhang@unileoben.ac.at

*Homepage*: Shuqin Zhang

*Mentors*: Peter Grabner, Peter Kirschenhofer, Benoît Loridant.

*Thesis Title*: "Geometry and topology of self-affine tiles and Rauzy fractals".

**First phase of the doctoral program:**-
**Milton Minervino**(Italy; October 2010–July 2014)

Personal homepage;*Email*: minervino@math.tugraz.at

*Mentors*: Peter Grabner, Peter Kirschenhofer.

*PhD Defense*: July 1, 2014.

*Referees*: V. Berthé (Paris), J. Thuswaldner, R. Tichy (TU Graz).

*Examiners*: J. Thuswaldner, R. Tichy (TU Graz).

### Project description

In the first phase of this project Milton Minervino was employed as PhD student. Within his thesis he studied dynamical and geometric properties of non-unimodular Pisot substitutions. Important connections of these substitutions with beta-numeration, cut and project schemes, as well as Rauzy fractals were established. In this setting the Rauzy fractals are compact subsets of certain open subrings of adele rings. It has been proved that they form tilings of these spaces under general conditions. This tiling property entails dynamical properties of infinite words defined in terms of the substitutions under consideration.

In the currently ongoing second phase of this project Shuqin Zhang is working on her PhD thesis. This thesis is concerned with topological properties of self-affine tiles. While there exist many results on the topology of 2-dimensional self-affine tiles, results in higher dimensions are rare and much harder to obtain. Nevertheless, in her thesis, Zhang succeeded to prove that under quite mild conditions a 3-dimensional self-affine tile has a boundary which is homeomorphic to a 2-sphere. Besides graph theory, a topological characterization of spheres that goes back to R. H. Bing as well as arguments from classical dimension theory are used. She is currently applying this new theory to 3-dimensional Rauzy fractals. Another part of her thesis deals with a systematic construction of space filling curves with certain optimal properties (like Hölder continuity) for self-similar sets.

Showcases for possible PhD themes can be found here.