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In this paper we give a geometric interpretation of the renormalizationalgorithm and of the continued fraction map that we introduced inarxiv:0905.0871 to give a characterization of symbolic sequences for linearflows in the regular octagon. We interpret this algorithm as renormalization onthe Teichmueller disk of the octagon and explain the relation with Teichmuellergeodesic flow. This connection is analogous to the classical relation betweenSturmian sequences, continued fractions and geodesic flow on the modularsurface. We use this connection to construct the natural extension and theinvariant measure for the continued fraction map. We also define anacceleration of the continued fraction map which has a finite invariantmeasure.

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