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Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle
TLDR
In this article, it was shown that the Forni bundle of the Hodge bundle is always flat and orthogonal to the tangent space of the moduli space of a curve.Abstract:
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M We show that the Forni bundle of N (ie the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N As a corollary, it follows that the Hodge bundle of N is semisimpleread more
Citations
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Every flat surface is Birkhoff and Oseledets generic in almost every direction
Jon Chaika,Alex Eskin +1 more
TL;DR: In this paper, it was shown that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodical theorem hold for every flat surface in almost every direction.
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Isolation, equidistribution, and orbit closures for the SL(2,R) action on Moduli space
TL;DR: In this article, the authors prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, analogous to the theory of unipotent flows.
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Cries and whispers in wind-tree forests
Vincent Delecroix,Anton Zorich +1 more
TL;DR: In this article, the authors studied the diffusion rate of the billiard in the plane endowed with symmetric periodic obstacles of a right-angled polygonal shape and proved that when the number of angles of a symmetric connected obstacle grows, diffusion rate tends to zero, thus answering a question of J.-C Yoccoz.
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Semisimplicity and rigidity of the Kontsevich-Zorich cocycle
TL;DR: In this paper, it was shown that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure, and that affine manifolds parametrize Jacobians with non-trivial endomorphisms.
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Zero Lyapunov exponents of the Hodge bundle
TL;DR: Forni and Trevi as mentioned in this paper showed that the Lyapunov spectrum of the Hodge bundle over the Teichm\"uller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the LyAPunov Spectrum.
References
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Book
Ergodic Theory and Semisimple Groups
TL;DR: In this paper, a generalization to p-adic groups and S-arithmetic groups is presented. But the generalization is not applicable to algebraic groups with Borel spaces.
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Extremal Lyapunov exponents: an invariance principle and applications
TL;DR: In this paper, it was shown that if the genus of the fiber is at least 2 then the Lyapunov exponents must be different from zero and vary continuously with the map, unless it is volume preserving conjugate to the automorphism itself.
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Invariant and stationary measures for the SL(2,R) action on Moduli space
Alex Eskin,Maryam Mirzakhani +1 more
TL;DR: In this article, it was shown that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold.
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Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices
TL;DR: In this article, a sufficient criterium for the existence of non-zero Lyapunov exponents for certain linear cocycles over hyperbolic transformations is proposed.
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