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The scenery flow for geometric structures on the torus: The linear setting. (English) Zbl 0993.37018

The authors define and study the scenery flow of the torus. The flow space is the union of all flat two-dimensional tori of area 1, each equipped with a quadratic differential of norm 1. This is a 5-dimensional space, and the flow consists in following points under an extremal deformation of the quadratic differential. The authors define associated horocyclic and translation flows. The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant 3-dimensional subset on which it is the suspension flow of that map. The authors give a simple definition of the scenery flow in terms of \(2\times 2\) matrices, using the group of affine maps of the plane. They prove that this flow is Anosov. They give then several descriptions of the flow and its cross sections, in particular using:
– symbolic dynamics (and in this framework, they relate the return map of the translation flow to the work of A. Vershik on \(p\)-adic transformations);
– a natural extension of two-dimensional continued fractions;
– induction on exchanges of three intervals;
– rescaling on pairs of transverse measured foliations on the torus, or the Teichmüller flow on the twice-punctured torus.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
30F60 Teichmüller theory for Riemann surfaces
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