A central limit theorem for random closed geodesics: Proof of the Chas–Li–Maskit conjecture

Gekhtman, Ilya; Taylor, Selwyn; Tiozzo, Giulio

doi:10.1016/j.aim.2019.106852

Cited by 11 publications

(11 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One might expect the classical CLT to hold in the setting of this paper, but none of these proof techniques are currently known to apply. We also mention an interesting recent realted result -an asymptotic central limit theorem for lengths of closed geodesics in hyperbolic surfaces was recently proved by Gekhtman, Taylor and Tiozzo [12].…”

Section: 2mentioning

confidence: 86%

Fluctuations of time averages around closed geodesics in non-positive curvature

Thompson,

Wang

2020

Preprint

View full text Add to dashboard Cite

We consider the geodesic flow for a rank one non-positive curvature closed manifold. We prove an asymptotic version of the Central Limit Theorem for families of measures constructed from regular closed geodesics converging to the Bowen-Margulis-Knieper measure of maximal entropy. The technique is based on generalizing ideas of Denker, Senti and Zhang, who proved this type of asymptotic Lindeberg Central Limit Theorem on periodic orbits for expansive maps with the specification property. We extend these techniques from the uniform to the non-uniform setting, and from discretetime to continuous-time. We consider Hölder observables subject only to the Lindeberg condition and a weak positive variance condition. Furthermore, if we assume a natural strengthened positive variance condition, the Lindeberg condition is always satisfied. We also generalize our results to dynamical arrays of Hölder observables, and to weighted periodic orbit measures which converge to a unique equilibrium state.

show abstract

Section: 2mentioning

confidence: 86%

Fluctuations of time averages around closed geodesics in non-positive curvature

Thompson,

Wang

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Note. The Chas-Li-Maskit conjecture has recently been proven in a preprint [10] by Gekhtman, Taylor, and Tiozzo. 1.3.…”

Section: Introductionmentioning

confidence: 90%

Conjugacy growth of commutators

Park

2019

Journal of Algebra

View full text Add to dashboard Cite

For the free group Fr on r > 1 generators (respectively, the free product G1 * G2 of two nontrivial finite groups G1 and G2), we obtain the asymptotic for the number of conjugacy classes of commutators in Fr (respectively, G1 * G2) with a given word length in a fixed set of free generators (respectively, the set of generators given by the nontrivial elements of G1 and G2). Our result is proven by using the classification of commutators in free groups and in free products by Wicks, and builds on the works of Rivin and Sharp, who asymptotically counted the conjugacy classes of commutator-subgroup elements in Fr with a given word length.

show abstract

“…Note. It has come to our attention that the statements of this paper are known-for example, they follow from Corollary 14.16 and Theorem 14.22 of [2]-and that the Chas-Li-Maskit Conjecture has recently been proven in [6]. and let • denote the operator norm on SL(V ).…”

Section: What Can We Say About the Relationship Between The Algebraic...mentioning

confidence: 99%

Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group

Park¹

2018

Preprint

View full text Add to dashboard Cite

Let S = Γ\H be a hyperbolic surface of finite topological type, such that the Fuchsian group Γ ≤ PSL2(R) is non-elementary, and consider any generating set S of Γ. When sampling by an n-step random walk in π1(S) ∼ = Γ with each step given by an element in S, the subset of this sampled set comprised of hyperbolic elements approaches full measure as n → ∞, and for this subset, the distribution of geometric lengths obeys a Law of Large Numbers, Central Limit Theorem, Large Deviations Principle, and Local Limit Theorem. We give a proof of this known theorem using Gromov's theorem on translation lengths of Gromov-hyperbolic groups.

show abstract

A central limit theorem for random closed geodesics: Proof of the Chas–Li–Maskit conjecture

Abstract: We prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface. In the special case of a hyperbolic pair of pants, this settles a conjecture of Chas-Li-Maskit. arXiv:1808.08422v2 [math.GT]

Cited by 11 publications

References 23 publications

Fluctuations of time averages around closed geodesics in non-positive curvature

Fluctuations of time averages around closed geodesics in non-positive curvature

Conjugacy growth of commutators

Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group

Contact Info

Product

Resources

About