Combinatorical and Algebraic Structures Seminar

Session details

Date: 5.3.2013
Speaker: Michal Kupsa, Ústav terorie informace a automatizace, AV ČR
Title: Codings of rotation II
Abstract: The number from the unit interval can be localized in many ways. The standard way is the decimal or binary expansion. In both cases, a long initial segment of the digital representation of the number determines a small subinterval the number lies in. The longer segment, the better precision and the smaller interval. The digits of the expansions can be achieved by the algorithm, which use repetitively the multiplication by the base and the integer part map. Another way how to localize the number is to use addition by an irrational number instead of multiplication by an integer. This approach leads to Sturmian sequences and codings of rotation. We show how the machinery works and present some classical results and one new result which sounds as follows: In the dynamics of a rotation of the unit circle by an irrational angle $\alpha\in(0,1)$, we study the evolution of partitions consisting of finite union of left-closed right-open intervals whose endpoints belong to the past trajectory of the point 0. We show that the refinements of these partitions eventually coincide with the refinements of a preimage of the Sturmian partition, which consists of two intervals $[0, 1 - \alpha)$ and $[1 - \alpha, 1)$. In particular, the refinements of the partitions eventually consist of connected sets, i.e., intervals. We reformulate this result in terms of Sturmian subshifts and the injectivity of the sliding block codes defined on them. This is folow-up of the seminary from 26.2.2012.

Return to index.