Beyond Sturmian sequences: coding linear trajectories in the regular octagon. (English) Zbl 1230.37021

Smillie, John; Ulcigrai, Corinna

Proc. Lond. Math. Soc. (3) 102, No. 2, 291-340 (2011).

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)].

Reviewer: Wolfgang Steiner (Sydney)

MSC:

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

Keywords:

Sturmian sequence; cutting sequence; billiards; Veech group; continued fraction; Farey map