### 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

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

**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 recent years, Thuswaldner worked together with Pierre Arnoux, Valerie Berthe, Milton Minervino, and Wolfgang Steiner in order to investigatie so-called S-adic words, see [Berthe et al. 2014; Berthe et al. 2016; Arnoux et al. 2018]. The aim of their work is to generalize a well-known relation between Sturmian words, continued fraction algorithms, and rotations on the torus to higher dimensions (this goes back to the work of Morse and Hedlund [1940] as well as to Rauzy). Their recent work involves the study of so-called S-adic words which are defined in terms of substitutions, generalizations of Rauzy fractals, generalized continued fraction algorithms, and rotation dynamics. In the course of their investigations they were able to set up a quite general conjugacy relation between shifts on S-adic words and rotations on tori thereby solving a problem of Arnoux and Rauzy going back to the early 1990ies. Thuswaldner is also interested in several aspects of numeration. A class of dynamical systems, so-called shift radix systems (see the survey [Kirschenhofer and Thuswaldner 2014]) are of importance in this context. In a series of papers J. Thuswaldner and coauthors study fundamental properties of shift radix systems and investigate their relations to numeration. Some years ago, he was able to set up a geometric theory for shift radix systems (cf. [Berthe et al. 2011]). In this theory a variety of fractal shapes occur, some of them are self-affine tiles, others are Rauzy fractals. However, also new kinds of fractals show up in this setting. The geometry as well as the tiling properties of these fractals reflect in arithmetic and dynamical properties of the underlying number systems and admit relations to the Pisot conjecture (see e.g. [Barge and Kwapisz 2006]). General results on these tiling properties have been achieved by defining so-called rational self-affine tiles, a natural generalization of the integral selfaffine tiles introduced by Lagarias and Wang [1996], to rational matrices (see [Steiner and Thuswaldner 2015]). The present project shall be carried out in interaction with the FWF funded projects P29910 and M2259. Jörg Thuswaldner is the leader (resp. supervisor) of these projects.