We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides hit. We give an explicit characterization of these cutting sequences. The cutting sequences for the square are the well-studied Sturmian sequences which can be analysed in terms of the continued fraction expansion of the slope. We introduce an analogous continued fraction algorithm which we use to connect the cutting sequence of a trajectory with its slope. Our continued fraction expansion of the slope gives an explicit sequence of substitution operations which generate the cutting sequences of trajectories with that slope. Our algorithm can be understood in terms of renormalization of the octagon translation surface by elements of the Veech group.
|Translated title of the contribution||Beyond Sturmian sequences: coding linear trajectories in the regular octagon|
|Pages (from-to)||291 - 340|
|Number of pages||50|
|Journal||Proceedings of the London Mathematical Society|
|Publication status||Published - Feb 2011|
Bibliographical notePublisher: London Mathematical Society
Other identifier: MathSciNet MR2769116