Beyond Sturmian sequences: symbolic sequences in regular 2n-gons

Sturmian sequences are sequences that code linear trajectories in a
square.It is well know that such sequences have minimal complexity and
that they are intimately related to the continued fraction map and the
dynamics of the geodesic flow on the modular surface.

We consider sequences that code linear trajectories in regular 2n-gons
(with opposite sides identified). We give complete characterization of (the
closure of) such symbolic sequences, in the same spirit of the
characterization of Sturmian sequences.

The sequences can be obtained through finitely many
"substitutions" associated to an appropriate additive continued fraction
map. The additive continued fraction algorithm exploits affine
diffeomorphisms in the Veech group and is related to the dynamics of the
Teichmuller geodesic flow on the corresponding Teichmuller disks of flat
surfaces.

This is a joint work with John Smillie.