The Scenery Flow for Geometric Structures on the Torus: The Linear Setting

Abstract

Abstract. We 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, on 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. We 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.We first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. We give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, we 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.

Initial powers of Sturmian sequences

“…The sequence (a k ) k≥1 turns out to be the sequence of partial quotients of the slope (defined as the density of the symbol 1), while (c k ) k≥1 is the sequence of digits in the arithmetic Ostrowski expansion of the intercept of the Sturmian sequence (see for instance [18,19,28,29,26,35,39,40] and the references in [10]). From this point of view, the characteristic (or standard ) Sturmian sequence of a particular slope is the one having c k = 0 for all k. This expansion of ω is just one of many possible expansions as an infinite composition of morphisms (see work of Arnoux [37], ArnouxFisher [5], Arnoux-Ferenczi-Hubert [4]). In each case these expansions are intimately linked to the Ostrowski numeration system.…”

Nonstationary Mixing and the Unique Ergodicity of Adic Transformations

“…Noting that a circle rotation is an exchange of two intervals, the proof follows as for other interval exchanges, as shown explicitly in [Fisa]; for the factor map, countably many pairs of points are identified (the endpoints of the Cantor set intervals). ⇤ For another proof, the factor map can be given an explicit arithmetical Ostrowsky formula as in [AF01] (there we use the dual substitution sequence). Related arithmetical formulas are studied in [GLT95] and a related Bratteli diagram (used for an a priori completely di↵erent purpose) appears in [ES80].…”

“…to irrational circle rotations, see Example 3) in the last section of [AF01]. We dedicate this paper to the memory of a very special person whom we have recently lost, Peter Bacas; he always pointed us toward the essentials, with such a warm heart and so much fire in his eyes.…”

Dynamical directions in numeration

“…For their metrical study, see [183,187]. For more on the connections between Ostrowski's numeration, word combinatorics, and particularly Sturmian words, see the survey [60], the sixth chapter in [272], and the very complete description of the scenery flow given in [31]. In the…”