THE SCENERY FLOWFOR GEOMETRIC STRUCTURES ON THE TORUS: THE LINEAR SETTING

The authors define the scenery flow of the torus.The flow space is the union of all flat 2-dimensional tori of area $1$with a marked direction (or equivalently, the union of all toriwith a quadratic differential of norm 1). This is a $5$-dimensionalspace, and the flow acts by following individual points under anextremal deformation of the quadratic differential. The authors defineassociated horocycle and translation flows; the latter preserve eachtorus and are the horizontal and vertical flows of the correspondingquadratic differential.The scenery flow projects to the geodesic flow on the modular surface,and admits, for each orientation preserving hyperbolictoral automorphism, an invariant $3$-dimensional subset on which it isthe suspension flow of that map.The authors first give a simple algebraicdefinition in terms of the group of affine maps of the plane, andprove that the flow is Anosov. They give an explicit formulafor the first-return map of the flow on convenient cross-sections. Then, inthe main part of the paper, the authors give several different modelsfor the flow and its cross-sections, in terms of:item{$bullet$} stacking and rescaling periodic tilings of the plane;item{$bullet$} symbolic dynamics: the natural extension of the recoding ofSturmian sequences, or the $S$-adic system generated by twosubstitutions;item{$bullet$} zooming and subdividing quasi-periodic tilings of the realline, or aperiodic quasicrystals of minimal complexity;item{$bullet$} the natural extension of two-dimensional continued fractions;item{$bullet$} induction on exchanges of three intervals;item{$bullet$} rescaling on pairs of transverse measure foliations on thetorus,or the Teichm"uller flow on the twice-punctured torus.