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Ergodic theory of interval exchange maps. (English) Zbl 1112.37003

The paper presents a detailed and unified review of results concerning interval exchange maps that have been developed over roughly the last thirty years. An introductory segment begins with definitions, examples and geometric interpretation. Basic tools and properties are described, such as Keane’s condition, Rauzy-Veech renormalization, and the Zorich transformation. Attention then is turned to translation surfaces, geodesic flow on them, Teichmüller flow, and their connections with interval exchange maps. The paper concludes by presenting one proof of Keane’s conjecture of the unique ergodicity of almost every interval exchange map (originally proved by Masur and by Veech), and of the existence of an ergodic, invariant probability measure for the Zorich renormalization operator.

MSC:

37A25 Ergodicity, mixing, rates of mixing
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
30F60 Teichmüller theory for Riemann surfaces
37F25 Renormalization of holomorphic dynamical systems
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
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