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Beyond Sturmian sequences: coding linear trajectories in the regular octagon. (English) Zbl 1230.37021
The symbolic coding of a linear trajectory in a regular $$2n$$-gon, where opposite sides are identified, keeps track of the sequence of sides hit by the trajectory. For $$n=2$$, the non-periodic cutting sequences are exactly the Sturmian sequences. In the present paper, non-periodic cutting sequences are characterized for the case $$n \geq 3$$ in terms of a derivation operator and a coherence condition. Here, derivation means that only sandwiched letters are kept, i.e., letters $$L$$ preceded and followed by the same letter $$L'$$.
Successive derivations and normalizations of the cutting sequence yield a $$2n$$-gon Farey expansion (or additive continued fraction expansion) of the angle of the linear trajectory. On the other hand, the continued fraction expansion gives a sequence of substitution operations that generate the cutting seqeunces of trajectories with that slope. In the case of the octagon, a direction has “terminating” Farey expansion if and only if it is in $$\mathbb{Q}(\sqrt{2})$$. This is similar to the case $$n=2$$, where terminating Farey expansions correspond to rational numbers. The factor complexity, i.e., the number of different words of length $$k$$, of a cutting sequence is bounded by $$(n-1) k + 1$$, and it is equal to $$(n-1) k + 1$$ when the direction is non-terminating.
The algorithm described by the authors can be understood in terms of renormalization of the $$2n$$-gon translation surface by elements of the Veech group; see also [the authors, Contemp. Math. 532, 29–65 (2010; Zbl 1222.37012)].

##### MSC:
 37B10 Symbolic dynamics 11A55 Continued fractions 37E35 Flows on surfaces
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