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The metric space of geodesic laminations on a surface. I. (English) Zbl 1063.57019

Authors’ abstract: We consider the space of geodesic laminations on a surface, endowed with the Hausdorff metric \(d_H\) and with a variation of this metric called the \(d_{\log}\) metric. We compute and/or estimate the Hausdorff dimensions of these two metrics. We also relate these two metrics to another metric which is combinatorially defined in terms of train tracks.

MSC:

57M99 General low-dimensional topology
37E35 Flows on surfaces
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References:

[1] J S Birman, C Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985) 217 · Zbl 0568.57006
[2] F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. \((2)\) 124 (1986) 71 · Zbl 0671.57008
[3] F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139 · Zbl 0653.32022
[4] F Bonahon, Transverse Hölder distributions for geodesic laminations, Topology 36 (1997) 103 · Zbl 0871.57027
[5] F Bonahon, Closed curves on surfaces, monograph in preparation
[6] F Bonahon, X Zhu, The metric space of geodesic laminations on a surface II: Small surfaces, Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry (2004) 509 · Zbl 1096.57016
[7] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge University Press (1988) · Zbl 0649.57008
[8] A Fathi, F Laudenbach, V Poénaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284
[9] K Falconer, Fractal geometry, John Wiley & Sons Ltd. (1990)
[10] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015
[11] A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press (1995) · Zbl 0878.58020
[12] H A Masur, Y N Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999) 103 · Zbl 0941.32012
[13] H A Masur, Y N Minsky, Geometry of the complex of curves II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902 · Zbl 0972.32011
[14] G Mess, Examples of Poincaré duality groups, Proc. Amer. Math. Soc. 110 (1990) 1145 · Zbl 0709.57025
[15] Y N Minsky, Combinatorial and geometrical aspects of hyperbolic 3-manifolds, London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press (2003) 3 · Zbl 1062.30053
[16] J R Munkres, Topology: a first course, Prentice-Hall (1975) · Zbl 0306.54001
[17] R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton University Press (1992) · Zbl 0765.57001
[18] W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1997-1980)
[19] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. \((\)N.S.\()\) 19 (1988) 417 · Zbl 0674.57008
[20] W P Thurston, Minimal stretch maps between hyperbolic surfaces
[21] X Zhu, Fractal dimensions of the space of geodesic laminations, PhD thesis, University of Southern California, Los Angeles (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.