Central limit theorem for the geodesic flow associated with a Kleinian group, case δ&gt;d/2

Enriquez, Nathanaël; Franchi, Jacques; Jan, Y Le

doi:10.1016/s0021-7824(00)01182-x

Cited by 9 publications

(10 citation statements)

References 14 publications

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“…Finally it is worth noticing again that, counter to the hyperbolic case of [2,3,4,6,7,11,12], the geodesic and Brownian asymptotic behaviors are here no longer the same, though comparable. The spiral windings of the geodesics about their projections on H 2 is mainly responsible for this feature.…”

Section: Introductionmentioning

confidence: 78%

“…Moreover the parameter a now appears in the limit law. This makes a noteworthy contrast with the hyperbolic case (see [2], [3], [6]). …”

Section: Lemma 7 As T → ∞ the Asymptotic Law Ofmentioning

confidence: 91%

“…The method for getting the geodesic result is based on a reduction to the Brownian behavior (calculated in Theorem 2), as in the series of articles [2,3,4,6,12]. There are however some noteworthy simplifications in comparison with the proofs in these articles, mainly due to the harmonicity of the integrated 1-forms, as in [11].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic windings over the trefoil knot

Franchi¹

2005

Rev. Mat. Iberoam.

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Consider the group G := P SL 2 (R) and its subgroups Γ := P SL 2 (Z) and Γ := DSL 2 (Z). G/Γ is a canonical realization (up to an homeomorphism) of the complement S 3 \ T of the trefoil knot T , and G/Γ is a canonical realization of the 6-fold branched cyclic cover of S 3 \ T , which has 3-dimensional cohomology of 1-forms.Putting natural left-invariant Riemannian metrics on G, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in G/Γ , describing thereby in an intrinsic way part of the asymptotic Brownian behavior in the fundamental group of the complement of the trefoil knot. A good basis of the cohomology of G/Γ , made of harmonic 1-forms, is calculated, and then the asymptotic Brownian behavior is obtained, by means of the joint asymptotic law of the integrals of the above basis along the Brownian paths.Finally the geodesics of G are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in G/Γ is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).

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Section: Introductionmentioning

confidence: 78%

“…Moreover the parameter a now appears in the limit law. This makes a noteworthy contrast with the hyperbolic case (see [2], [3], [6]). …”

Section: Lemma 7 As T → ∞ the Asymptotic Law Ofmentioning

confidence: 91%

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Asymptotic windings over the trefoil knot

Franchi¹

2005

Rev. Mat. Iberoam.

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“…. , N , let 4 k=0 P T −t k (|∇ (k) df | p ), for some p. Thus there are numbers c, q such that for all k, 7) for some constant C(T ) and some functionγ in B V,0 . If f ∈ BC 4 , it is easy to see that there is a function g ∈ B V,0 , not depending on f , s.t.…”

Section: Proofmentioning

confidence: 99%

Limits of random differential equations on manifolds

2015

Probab. Theory Relat. Fields

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“…See Sinai [30], Ratner [28]; see Guivarch and Le Jan [12] and Enriquez, Franchi and Le Jan [8] for further developments. See also Helland [15] and Kipnis and Varadhan [19].…”

mentioning

confidence: 99%

Random perturbation to the geodesic equation

Li¹

2016

Ann. Probab.

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We study random “perturbation” to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm 11. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by 8n(n−1)8n(n−1) where nn is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion

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Central limit theorem for the geodesic flow associated with a Kleinian group, case δ>d/2

Cited by 9 publications

References 14 publications

Asymptotic windings over the trefoil knot

Asymptotic windings over the trefoil knot

Limits of random differential equations on manifolds

Random perturbation to the geodesic equation

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