Weakly Mixing Polygonal Billiards

Chaika, Jon; Forni, Giovanni

doi:10.48550/arxiv.2003.00890

Cited by 1 publication

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“…The strategy of the argument, based on the so-called freezing argument from [CF20], consists in deriving from the above Lipshitz property the existence of a proper SL(2, R)-invariant subbundle of P(H), thereby contradicting the strong irreducibility assumption. Recall that the function Ψ 1 is given by the formula (see formula (2.6.2))…”

Section: Proofs Of Main Theoremsmentioning

confidence: 99%

A Central Limit Theorem for the Kontsevich-Zorich Cocycle

Al-Saqban¹,

Forni²

2022

Preprint

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In this note, we show that a central limit theorem holds for exterior powers of the Kontsevich-Zorich (KZ) cocycle. In particular, we show that, under the hypothesis that the top Lyapunov exponent on the exterior power is simple, a central limit theorem holds for the lift of the (leafwise) hyperbolic Brownian motion to any strongly irreducible, symplectic, SL(2, R)-invariant subbundle, that is moreover symplectic-orthogonal to the so-called tautological subbundle. We then show that this implies that a central limit theorem holds for the lift of the Teichmüller geodesic flow to the same bundle.For the random cocycle over the hyperbolic Brownian motion, we are able to prove under the same hypotheses that the variance of the top exponent is strictly positive. For the deterministic cocycle over the Teichmüller geodesic flow we can prove that the variance is strictly positive only for the top exponent of the first exterior power (the KZ cocycle itself) under the hypothesis that its Lyapunov spectrum is simple.

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Section: Proofs Of Main Theoremsmentioning

confidence: 99%