Combinatorical and Algebraic Structures Seminar

Session details

Date: 17.3.2006
Speaker: Wolfgang Steiner, LIAFA, Université Paris VII
Title: Periodicity of discrete rotations and domain exchange
Abstract: (spoluautoři: S. Akiyama, H. Brunotte and A. Pethoe)
The points $(a_n,a_{n-1})$ of a sequence defined by the recurrence $a_{n+1}=-\lambda a_n-a_{n-1}$ lie on an ellipse if $-2<\lambda<2$. In this talk, we discuss an approximation of this sequence with integer points defined by $0\le a_{n+1}+\lambda a_n+a_{n-1}<1$. This is a special case of the shift radix systems defined by Akiyama et al. It is conjectured that all these sequences are periodic.
If the rotation angle is a rational multiple of $\pi$, then we can associate a domain exchange in $[0,1)^{2k}$, where $k$ is the algebraic degree of $\lambda$. Up to now, only the quadratic numbers ($\lambda=(\pm 1\pm\sqrt{5})/2, \pm\sqrt{2}, \pm\sqrt{3}$) have been studied in detail. For these $\lambda$, the domain exchange has a self similar structure. We can prove the periodicity of all sequences and calculate the possible period lengths.

Return to index.