  In this paper we give a geometric interpretation of the renormalization
algorithm and of the continued fraction map that we introduced in
arxiv:0905.0871 to give a characterization of symbolic sequences for linear
flows in the regular octagon. We interpret this algorithm as renormalization on
the Teichmueller disk of the octagon and explain the relation with Teichmueller
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.
