Geodesic flow on the Teichm\"uller 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 our previous paper (Beyond Sturmian Sequences: coding linear trajectories in the regular octagon, Proceedings London Mathematical Society) to give a characterization of symbolic sequences for linear flows in the regular octagon. We interpret this algorithm as renormalization on the Teichm¨uller disk of the octagon and explain the relation with Teichm¨uller 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.