Variations on a theorem of Birman and Series
[Variations sur un théorème de Birman et Series]
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 171-194.

Soient Σ une surface hyperbolique et f: + + une fonction monotone. Nous étudions l’adherence dans le fibré projectif tangent PTΣ de l’ensemble des géodésiques γ telles que i(γ,γ)f( Σ (γ)). En particulier nous montrons que si f est non bornée et sous-linéaire alors la dimension de Hausdorff de cet ensemble est strictement entre 1 et 3.

Suppose that Σ is a hyperbolic surface and f: + + a monotonic function. We study the closure in the projective tangent bundle PTΣ of the set of all geodesics γ satisfying I(γ,γ)f( Σ (γ)). For instance we prove that if f is unbounded and sublinear then this set has Hausdorff dimension strictly bounded between 1 and 3.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3156
Classification : 30F10,  30F60
Mots clés : géodesiques, surfaces hyperboliques, auto-intersection, dimension de Hausdorff
@article{AIF_2018__68_1_171_0,
     author = {Lenzhen, Anna and Souto, Juan},
     title = {Variations on a theorem of Birman and Series},
     journal = {Annales de l'Institut Fourier},
     pages = {171--194},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3156},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3156/}
}
Lenzhen, Anna; Souto, Juan. Variations on a theorem of Birman and Series. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 171-194. doi : 10.5802/aif.3156. http://archive.numdam.org/articles/10.5802/aif.3156/

[1] Aramayona, Javier; Leininger, Christopher Hyperbolic structures on surfaces and geodesic currents (to appear)

[2] Benson, Brian The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces (2016) (https://arxiv.org/abs/1509.08993)

[3] Birman, Joan S.; Series, Caroline Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology, Volume 24 (1985), pp. 217-225 | Article | Zbl 0568.57006

[4] Bonahon, Francis Bouts des variétés hyperboliques de dimension 3, Ann. Math., Volume 124 (1986), pp. 71-158 | Article | Zbl 0671.57008

[5] Bonahon, Francis The geometry of Teichmüller space via geodesic currents, Invent. Math., Volume 92 (1988) no. 1, pp. 39-162 | Article | Zbl 0653.32022

[6] Buser, Peter A note on the isoperimetric constant, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982), pp. 213-230 | Article | Zbl 0501.53030

[7] Buser, Peter; Sarnak, Peter C. On the period matrix of a Riemann surface of large genus, Invent. Math., Volume 117 (1994) no. 1, pp. 27-56 | Article | Zbl 0814.14033

[8] Casson, Andrew J.; Bleiler, Steven A. Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, 9, Cambridge University Press, 1988, 105 pages | Zbl 0649.57008

[9] Cheeger, Jeff A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969), Princeton University Press, 1970, pp. 195-199 | Zbl 0212.44903

[10] Patterson, Samuel James The limit set of a Fuchsian group, Acta Math., Volume 136 (1976), pp. 241-273 | Article | Zbl 0336.30005

[11] Sapir, Jenya Non-simple geodesics on surfaces (2014) (Ph. D. Thesis)

[12] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions, Publ. Math., Inst. Hautes Étud. Sci., Volume 50 (1979), pp. 171-202 | Article | Zbl 0439.30034