Variations on a central limit theorem in infinite ergodic theory

Thomine, Damien

doi:10.1017/etds.2013.114

Cited by 7 publications

(26 citation statements)

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“…Finally, let us mention that the geodesic flow on periodic manifolds is also used to study the horocycle flow on the same manifolds [6,41,42,43]. Limit theorems for observables with integral zero have already been obtained in this context [64], but the improvement we get with Theorem 1.11 translates into a limit theorem which is valid for a wider class of observables. Instead of having compact support, the observables need only to decay polynomially fast at infinity.…”

Section: 42mentioning

confidence: 98%

“…) are asymptoticaly independent [63, Theorem 1.7], and we have a generalized central limit theorem [64,Corollary 6.9], which has the following form when d = 1:…”

Section: )mentioning

confidence: 99%

“…Let ε > 0. Let δ > 0 and q > 2 be small enough such that: We are now ready to apply again [64,Theorem 6.8]. Let f : A → R be such that:…”

Section: Tightnessmentioning

confidence: 99%

“…Let ϕ A×{0} be the first return time to A × {0} for the geodesic flow, and ϕ A×{0} the first return time to A × {0} for T . The proof then proceeds as in [64], with the same weakened criterion: we only need to check that, for some δ > 0, sup…”

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confidence: 99%

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Potential kernel, hitting probabilities and distributional asymptotics

Pène¹,

Thomine²

2019

Ergod. Th. Dynam. Sys.

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Z d -extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green-Kubo's formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve in turn the asumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on abelian covers of compact manifolds of negative curvature are discussed. arXiv:1702.06625v2 [math.DS] 16 May 2017 POTENTIAL KERNEL, HITTING PROBABILITIES AND DISTRIBUTIONAL ASYMPTOTICS 2 observables need only to decay polynomially at infinity, instead of having bounded support. We apply it to the geodesic flow on abelian covers of compact manifolds with negative curvature.This article is organized as follow. We present our setting and our results in Section 1, as well as our applications to Lorentz gases (Sub-subsection 1.4.1) and to geodesic flows (Sub-subsection 1.4.2). In Section 2 we present our spectral assumptions, and prove Theorem 1.4 using the method of moments. In Section 3 we prove Theorems 1.7 and 1.11, and in Section 4 the two applications mentioned above. We discuss Green-Kubo's formula in the Appendix.

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Section: 42mentioning

confidence: 98%

“…) are asymptoticaly independent [63, Theorem 1.7], and we have a generalized central limit theorem [64,Corollary 6.9], which has the following form when d = 1:…”

Section: )mentioning

confidence: 99%

“…Let ε > 0. Let δ > 0 and q > 2 be small enough such that: We are now ready to apply again [64,Theorem 6.8]. Let f : A → R be such that:…”

Section: Tightnessmentioning

confidence: 99%

mentioning

confidence: 99%

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Potential kernel, hitting probabilities and distributional asymptotics

Pène¹,

Thomine²

2019

Ergod. Th. Dynam. Sys.

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“…In [56], by researching on the monotonic properties of the convergence factor established by applying the Fourier transform to the error functions, Wu and Wu gave a realiable choice of the Robin parameter in the nonoverlapping case and numerical results, which mean that the analyzed Robin parameter results in satisfactory convergence rate. In [48], Thomine extended the corresponding result to larger classes of observables, with milder smoothness conditions, and to larger classes of dynamical systems, which may no longer be mixing. A special emphasis is given by him to continuous time systems: semi-flows, flows, and Z d 1 -extensions of flows, where d 1 0.…”

Section: Introductionmentioning

confidence: 94%

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Li¹,

Lu²,

Yu³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, for a continuous semi-flow θ on a compact metric space E with the asymptotic average-shadowing property (AASP), we show that if the almost periodic points of θ are dense in E then θ is multi-sensitive and syndetically sensitive. Also, we show that if θ is a Lyapunov stable semi-flow with the AASP, then the space E is trivial. Consequently, a Lyapunov stable semi-flow with the AASP is minimal. Furthermore, we prove that for a syndetically transitive continuous semi-flow on a compact metric space, sensitivity is equivalent to syndetical sensitivity. As an application, we show that for a continuous semi-flow θ on a compact metric space E with the AASP, if the almost periodic points of ϕ are dense in E then θ is syndetically sensitive. Moreover, we prove that for any continuous semi-flow θ on a compact metric space, it has the AASP if and only if so does its inverse limit ( E, θ), and if only if so does its lifting continuous semi-flow ( E, θ). Also, an example which contains two numerical experiments is given. Our results extend some corresponding and existing ones.

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Local time and first return time for periodic semi-flows

Thomine

2016

Isr. J. Math.

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Variations on a central limit theorem in infinite ergodic theory

Cited by 7 publications

References 15 publications

Potential kernel, hitting probabilities and distributional asymptotics

Potential kernel, hitting probabilities and distributional asymptotics

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Local time and first return time for periodic semi-flows

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