Local geometry of random geodesics on negatively curved surfaces

Athreya, Jayadev; Lalley, Steve; Sapir, Jenya; Wroten, Matthew

Annales Henri Lebesgue, Tome 4 (2021), pp. 187-226.

Nous montrons que la tessellation d’une surface compacte de courbure strictement négative induite par un long segment géodésique aléatoire ressemble localement à un processus de Poisson en droites, après rééchelonnement. Ceci implique que les statistiques globales de la tessellation (par exemple, la proportion de triangles) convergent vers celles du processus de Poisson en droites limite.

We show that the tessellation of a compact, negatively curved surface induced by a long random geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.

Classification : 37D40, 37E35, 37B10
Mots clés : self-intersection, random tessellation, geodesic, hyperbolic surface, Poisson line process

[AF82] Adler, Roy L.; Flatto, Leopold Cross section map for the geodesic flow on the modular surface, Conference in modern analysis and probability (New Haven, Conn., 1982) (Contemporary Mathematics), Volume 26, American Mathematical Society (1982), pp. 9-24 | DOI | Zbl

[Bil68] Billingsley, Patrick Convergence of probability measures, John Wiley & Sons, 1968 | Zbl

[Bow72] Bowen, Rufus The equidistribution of closed geodesics, Am. J. Math., Volume 94 (1972), pp. 413-423 | DOI | MR | Zbl

[Bow73] Bowen, Rufus Symbolic dynamics for hyperbolic flows, Am. J. Math., Volume 95 (1973), pp. 429-460 | DOI | MR | Zbl

[Bow75] Bowen, Rufus Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, 470, Springer, 1975 | MR | Zbl

[BS79] Bowen, Rufus; Series, Caroline Markov maps associated with Fuchsian groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 50 (1979), pp. 153-170 | DOI | Numdam | Zbl

[BW72] Bowen, Rufus; Walters, Peter Expansive one-parameter flows, J. Differ. Equations, Volume 12 (1972), pp. 180-193 | DOI | MR | Zbl

[Cal53] Calderon, Alberto P. A general ergodic theorem, Ann. Math., Volume 58 (1953), pp. 182-191 | DOI | MR | Zbl

[Cal03] Calka, Pierre An explicit expression for the distribution of the number of sides of the typical Poisson–Voronoi cell, Adv. Appl. Probab., Volume 35 (2003) no. 4, pp. 863-870 | DOI | MR | Zbl

[CFF02] Comets, Francis M.; Fernández, Roberto; Ferrari, Pablo A. Processes with long memory: regenerative construction and perfect simulation, Ann. Appl. Probab., Volume 12 (2002) no. 3, pp. 921-943 | MR | Zbl

[Dol04] Dolgopyat, Dmitry Limit theorems for partially hyperbolic systems, Trans. Am. Math. Soc., Volume 356 (2004) no. 4, pp. 1637-1689 | DOI | MR | Zbl

[DZ98] Dembo, Amir; Zeitouni, Ofer Large deviations techniques and applications, Applications of Mathematics, 38, Springer, 1998 | MR | Zbl

[EJM91] Epstein, Charles; Jorgensen, Troels; McMullen, Curt Multiple intersections on negatively curved surfaces, J. Differ. Geom., Volume 33 (1991) no. 1, pp. 253-261 | DOI | MR | Zbl

[Kin93] Kingman, John F. C. Poisson processes, Oxford Studies in Probability; Oxford Science Publications, 3, Clarendon Press; Oxford University Press, 1993 | MR | Zbl

[KM99] Kleinbock, Dmitry Y.; Margulis, Grigoriĭ A. Logarithm laws for flows on homogeneous spaces, Invent. Math., Volume 138 (1999) no. 3, pp. 451-494 | DOI | MR | Zbl

[Lal86] Lalley, Steven P. Regenerative representation for one-dimensional Gibbs states, Ann. Probab., Volume 14 (1986) no. 4, pp. 1262-1271 | DOI | MR | Zbl

[Lal87] Lalley, Steven P. Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math., Volume 8 (1987) no. 2, pp. 154-193 | DOI | MR | Zbl

[Lal89] Lalley, Steven P. Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J., Volume 58 (1989) no. 3, pp. 795-821 | MR | Zbl

[Lal96] Lalley, Steven P. Self-intersections of closed geodesics on a negatively curved surface: statistical regularities, Convergence in ergodic theory and probability (Columbus, OH, 1993) (Ohio State University Mathematical Research Institute Publications), Volume 5, Walter de Gruyter, 1996, pp. 263-272 | MR | Zbl

[Lal14] Lalley, Steven P. Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces, Duke Math. J., Volume 163 (2014) no. 6, pp. 1191-1261 | DOI | MR | Zbl

[LC60] Le Cam, Lucien An approximation theorem for the Poisson binomial distribution, Pac. J. Math., Volume 10 (1960), pp. 1181-1197 | DOI | MR | Zbl

[LNP19] Li, Jialun; Naud, Frédéric; Pan, Wenyu Kleinian Schottky groups, Patterson–Sullivan measures, and Fourier decay, with an appendix on stationarity of Patterson–Sullivan measures (2019) (https://arxiv.org/abs/1902.01103, to appear in Duke Mathematical Journal)

[Mau06] Maucourant, François Dynamical Borel–Cantelli lemma for hyperbolic spaces, Isr. J. Math., Volume 152 (2006), pp. 143-155 | DOI | MR | Zbl

[Mil64a] Miles, Roger E. Random polygons determined by random lines in a plane, Proc. Natl. Acad. Sci. USA, Volume 52 (1964), pp. 901-907 | DOI | MR

[Mil64b] Miles, Roger E. Random polygons determined by random lines in a plane. II, Proc. Natl. Acad. Sci. USA, Volume 52 (1964), pp. 1157-1160 | DOI | MR | Zbl

[PPS15] Paulin, Frédéric; Pollicott, Mark; Schapira, Barbara Equilibrium states in negative curvature, Astérisque, 373, Société Mathématique de France, 2015 | Zbl

[Rat73] Ratner, Marina Markov partitions for Anosov flows on $n$-dimensional manifolds, Isr. J. Math., Volume 15 (1973), pp. 92-114 | DOI | MR | Zbl

[San04] Santaló, Luis A. Integral geometry and geometric probability, Cambridge Mathematical Library, Cambridge University Press, 2004 (with a foreword by Mark Kac) | Zbl

[Ser81] Series, Caroline Symbolic dynamics for geodesic flows, Acta Math., Volume 146 (1981) no. 1-2, pp. 103-128 | DOI | MR | Zbl

[Ser85] Series, Caroline The modular surface and continued fractions, J. Lond. Math. Soc., Volume 31 (1985) no. 1, pp. 69-80 | DOI | MR | Zbl

[SKM87] Stoyan, Dietrich K.; Kendall, Wilfrid S.; Mecke, Joseph Stochastic geometry and its applications, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1987 (With a foreword by D. G. Kendall.) | Zbl

[Sul82] Sullivan, Dennis Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., Volume 149 (1982) no. 3, pp. 3-4 | MR | Zbl

[SY94] Schoen, Richard; Yau, Shing-Tung Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, 1, International Press, 1994 (Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso.) | Zbl