Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

Filip, Simion

doi:10.1007/s00222-015-0643-3

Cited by 41 publications

(35 citation statements)

References 43 publications

(87 reference statements)

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“…As the last set is symmetric, the Lyapunov exponents of ((A p ) −1 ) tr and A p are the same. Moreover, ((A p ) −1 ) tr meets all properties of A p used in the current part of the proof, more precisely, ((A p ) −1 ) tr is the composition of A with a group endomorphism so that the results of [16] can be applied. It follows that for every 0 < ε < 1 there exist λ > 0 and L ∈ N such that for a.e.…”

Section: (113)mentioning

confidence: 80%

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Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

Frączek¹,

Shi²,

Ulcigrai³

2018

Journal of Modern Dynamics

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In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices ASL 2 (R)/ASL 2 (Z), we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum H(1, 1) of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragović and Radnović, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Frączek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.

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Section: (113)mentioning

confidence: 80%

“…Filip in [16] showed that the subbundles V i → M, 1 ≤ i ≤ k locally vary polynomially in the period coordinates. In view of [12], one can deduce that V i → M, 1 ≤ i ≤ k are indeed locally constant whenever the space W ⊂ H 1 (M, R) satisfies the assumptions of Theorem 2.7, as we now explain.…”

Section: (113)mentioning

confidence: 99%

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

Frączek¹,

Shi²,

Ulcigrai³

2018

Journal of Modern Dynamics

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“…These works are concerned with formulas for exponents, spectral gap and non-triviality of exponents, as well as applications to dynamics on translation surfaces. Later, in [8] it was proved that the KZ cocycle is semisimple and its decomposition respects the Hodge structure. Using this property, in [9], an analysis of possible groups appearing in the Zariski closure of the monodromy (or the algebraic hull) was done.…”

Section: Introductionmentioning

confidence: 99%

Quaternionic covers and monodromy of the Kontsevich–Zorich cocycle in orthogonal groups

2017

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ABSTRACT. We give an example of a Teichmüller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group SO * (6) in its standard representation; up to finite index, this is the same as SU (3, 1) in its second exterior power representation.The example is constructed using origamis (i.e. square-tiled surfaces). It can be generalized to give monodromy inside the group SO * (2n) for all n, but in the general case the monodromy might split further inside the group.Also, we take the opportunity to compute the multiplicities of representations in the (0,1) part of the cohomology of regular origamis, answering a question of Matheus-Yoccoz-Zmiaikou.

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“…Despite the fact that Teichmüller space is in a sense completely inhomogeneous [70], analogies to the dynamics of homogeneous spaces provide insight that has allowed for astounding progress. A decade after McMullen's [54] treatment of the genus 2 case, Eskin and Mirzakhani [17] show that finite ergodic SL 2 (R)-invariant measures are of Lebesgue class and supported on affine varieties; Eskin, Mirzakhani, and Mohammadi [18] show that the closures of SL 2 (R)-orbits are affine manifolds; and Filip [21,22] shows that these closures are algebraic varieties defined over number fields, thus generalizing the aforementioned results for Teichmüller curves. For more on these results and the flurry of work they have inspired see [82].…”

Section: Teichmüller Dynamics and The Hodge Bundlementioning

confidence: 92%