Geodesic flow on the Teichmuller disk of the regular octagon, cutting sequences and octagon continued fractions maps

In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in [15] to give a characterization of symbolic sequences for linear flows in the regular octagon. We interpret this algorithm as renormalization on the Teichmuller disk of the octagon and explain the relation with Teichmuller geodesic flow. This connection is analogous to the classical relation between Sturmian sequences, continued fractions and geodesic flow on the modular surface. We use this connection to construct the natural extension and the invariant measure for the continued fraction map. We also define an acceleration of the continued fraction map which has a finite invariant measure.