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Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. (English) Zbl 1143.37001
The paper contains a proof of the Zorich conjecture that the Lyapunov exponents of the Zorich cocyle on each Rauzy class are distinct. The Zorich cocyle is a linear cocycle over a renormalization map on the space of interval exchange transformations. The larger part of the paper is devoted to showing that the Zorich cocycle on any Rauzy class satisfies the hypotheses of a result that gives sufficient conditions for the Lyapunov exponents of locally constant cocycles to be distinct. (This latter result is given in full generality in a separate paper by the same authors [Port. Math. (N.S.) 64, No. 3, 311–376 (2007; Zbl 1137.37001)], while an appendix of the paper under review contains a proof of a version sufficient to complete the proof of the main theorem.) The argument is framed in terms of properties, called “twisting” and “pinching”, of symplectic actions of monoids. These properties are shown to imply simplicity of the monoid action, which is the main hypothesis of the result referred to above. The paper gives combinatorial analysis of Rauzy classes and the relations between them, showing that the twisting and pinching properties are present in this context.

MSC:
37A25 Ergodicity, mixing, rates of mixing
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
28D05 Measure-preserving transformations
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