THE SCENERY FLOW FOR GEOMETRIC STRUCTURES ON THE TORUS: THE LINEAR SETTING

Abstract

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 marketed direction (or equivalently, the union of all tori with a quadratic differential of norm 1). This is a 5-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential.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 first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of:• stacking and rescaling periodic tilings of the plane;• symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the S-adic system generated by two substitutions;• zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity;• the natural extension of two-dimensional continued fractions;• induction on exchanges of three intervals;• rescaling on pairs of transverse measure foliations on the torus, or the Teichmüller flow on the twice-punctured torus.