Distribution of intersection lengths of a random geodesic with a geodesic lamination
Open accessAbstract
We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distribution are related to the Riemann zeta function.
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Lemma Appendix A.2: For any hyperbolic surface Σp with marking (X,f), the stated identity holds, where χ(Σp) = 2(1-g) is the Euler characteristic; this is a topological invariant.
“Lemma Appendix A.2. [26, 27] For any hyperbolic surface Σ p with the marking ( X,f ) , we have the following result”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [26]
Section 5 (page unavailable) gives an identity for a closed hyperbolic surface involving its Euler characteristic: the number of complementary ideal triangles is 2|χ(S)|. The cited paper does not state the claim’s unspecified identity for a surface with marking (X,f), nor does it give χ(Σp)=2(1−g) or explicitly describe χ as a topological invariant. (LLM, conf=0.72)
Experimental — accuracy not yet measuredLemma Appendix A.1: The map (X,f) → L_X is a proper embedding of TΣp into the space of currents Curr(Σp) given by the intersection number.
“Lemma Appendix A.1. [26, 27] The map ( X,f ) → ˆ L X is a proper embedding of T Σ p into the space of currents Curr (Σ p ) given by the intersection number i that is, for all closed curves α in Σ p ,”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [26]
Section 2, “Geodesic laminations and geodesic currents” (page unavailable), identifies the Liouville measure of a closed hyperbolic surface as a geodesic current. It does not state that the map from Teichmüller space to currents is a proper embedding or characterize that map by intersection numbers. (LLM, conf=0.78)
Experimental — accuracy not yet measuredA sequence diverging in Teichmüller space converges in the space of projective measured laminations (PML), which is a compact subset of the space of geodesic currents.
“Here we show that a sequence diverging in Teichmüller space converges in the space of projective measured laminations ( PML ), which is a compact subset of the space of geodesic currents.”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [26]
Section 2, “Geodesic laminations and geodesic currents” (page not provided), defines geodesic currents but does not discuss PML or Teichmüller space. Searches for Teichmüller space, PML, and compactness found no passage addressing whether a diverging sequence converges in PML. (LLM, conf=0.75)
Experimental — accuracy not yet measured
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Cites
No references could be resolved yet.
Research works printed but not linked: 5
- Francis Bonahon. The geometry of Teichmüller space via geodesic currents. Invent. Math., 92(1):139-162, 1988.
- I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaȋ. Ergodic Theory, volume 245 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiȋ.
- Peter J. Nicholls. The Ergodic Theory of Discrete Groups, volume 143 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1989.
- Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.
- Stephen Wolfram. Mathematica: A System for Doing Mathematics by Computer. Addison- Wesley, 1991.
Cited by
Sorted by citations
Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space
Mondal, P. (2023) · Journal of Mathematical Physics · doi:10.1063/5.0136631 · Cited by 1 · see citing works
- Authors
- BRIDGEMAN, M., & DUMAS, D.
- Affiliations
- Boston College (US)
- Brown University (US)
- Venue
- Ergodic Theory and Dynamical Systems
- Publisher
- Cambridge University Press (CUP)
- Published
- 2007-05-03
- Volume
- 27
- Issue
- 4
- Pages
- 1055-1072
- ISSN
- 0143-3857 (ISSN-L)
- 1469-4417
- OpenAlex
- W2125847302
- Cited by
- 18
- References
- 0
- Citations by year
- 2023: 1 · 2022: 1 · 2020: 1 · 2017: 1 · 2016: 1 · 2015: 1 · 2014: 2 · 2013: 1 · 2012: 1
- FWCI
- 2.92
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Ergodic Theory and Dynamical Systems (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- geodesic laminations, Riemann zeta function, moments of distributions, hyperbolic surfaces, ergodic theory
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