Abstract
This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'' (from 4 to 16 July, 2011).   In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmuller and moduli space of translation surfaces, the Teichmuller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmuller geodesic flow. We sketch two applications of the ergodic properties of the Teichmuller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmuller flow and the Kontsevich--Zorich cocycle work as renormalization dynamics for interval exchange transformations and translation flows.   In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmuller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).
Highlights
TO TEICHMU LLER THEORY AND ITS APPLICATIONS TO DYNAMICS OF INTERVAL EXCHANGE TRANSFORMATIONS, FLOWSON SURFACES AND BILLIARDSGIOVANNI FORNI AND CARLOS MATHEUS AbstractThis text is an expanded version of the lecture notes of a minicourse delivered by the authors in the Bedlewo school “Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory”.In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmuller and moduli space of translation surfaces, the Teichmuller flow and the SL(2, R)-action on these moduli spaces and the Kontsevich–Zorich cocycle over the Teichmuller geodesic flow
We study some examples of measures for which the ergodic properties of the Kontsevich–Zorich cocycle are very different from the case of Masur– Veech measures
The long-term goal of these lecture notes is the study of the so-called Teichmuller geodesic flow and its noble cousin the Kontsevich–Zorich cocycle, and some of its applications to interval exchange transformations, translation flows and billiards
Summary
The long-term goal of these lecture notes is the study of the so-called Teichmuller geodesic flow and its noble cousin the Kontsevich–Zorich cocycle, and some of its applications to interval exchange transformations, translation flows and billiards. Teichmuller’s theorem says that extremal maps f : M0 → M1 (i.e., deformations of Riemann surface structures) can be understood in terms of half-translation structures: it suffices to expand (resp., contract) the corresponding horizontal (resp., vertical) foliation on M0 by a constant factor equal to ed(M0,M1) in order to get a horizontal (resp., vertical) foliation of a half-translation structure compatible with the Riemann surface structure of M1 This provides a simple way to describe the Teichmuller geodesic flow in terms of half-translation structures. Given a half-translation structure {(Ui, φi)} (where φi : Ui → C) on a Riemann surface S, one can construct a quadratic differential q on S by pulling back the quadratic differential dz on C through the map φi on every Ui ⊂ S: this procedure leads to a well-defined global quadratic differential on S because we are assuming that the changes of coordinates (outside the neighborhoods of finitely many points) have the form z → ±z + c. We note that the basic difference between the Teichmuller metric and the Weil-Petersson metric is the following: as we already indicated, the Teichmuller metric is related to flat (half-translation) structures, while the Weil-Petersson metric can be better understood in terms of hyperbolic structures
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