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ENTROPY RIGIDITY FOR METRIC SPACES

  • Lim, Seon-Hee (Department of Mathematics, Seoul National University)
  • Received : 2012.01.31
  • Accepted : 2012.02.20
  • Published : 2012.02.28

Abstract

This is a survey on the volume entropy and its rigidity of various metric spaces. This survey is aimed to summarize recent results as well as remaining open questions and possible directions on this subject.

Keywords

References

  1. G. Besson, G. Courtois & S. Gallot: Entropies et rigidites des espaces localement symetriques de courbure strictement negative. Geom. Funct. Anal. 5 (1995), 731-799. https://doi.org/10.1007/BF01897050
  2. G. Besson: Volumes and entropies. Riemannian geometry (Waterloo, ON, 1993). 1-22, Fields Inst. Monogr., 4, Amer. Math. Soc., Providence, RI, 1996.
  3. G. Berck, A. Bernig & C. Vernicos: Volume entropy of Hilbert geometries. Pacific J. Math. 245 (2010), no. 2, 201-225. https://doi.org/10.2140/pjm.2010.245.201
  4. Y. Benoist: Convexes hyperboliques et fonctions quasisymetriques. (French. English summary) [Hyperbolic convex sets and quasisymmetric functions] Publ. Math. Inst. Hautes Etudes Sci. No. 97 (2003), 181-237.
  5. A. Bernig: Hilbert geometry of polytopes. Arch. Math. (Basel) 92 (2009), no. 4, 314- 324. https://doi.org/10.1007/s00013-009-3142-1
  6. L. Bessiµeres, A. Parreau & B. Remy: Geometries µa courbure negative ou nulle, groupes discrets et rigidites. Seminaires & Congres [Seminars and Congresses], 18, Lecture notes from the Summer School held at the Institute Fourier, Grenoble, 2004, Societe Mathematique de France, 2009.
  7. J. Boland & F. Newberger: Minimla entropy rigidity for Finsler manifolds of negative flag curvature. Ergod. Th. and Dynam. Sys. 21 (2001), no. 1, 13-23.
  8. M. Bourdon & H. Pajot: Quasi-conformal geometry and hyperbolic geometry. Rigidity in dynamics and geometry (Cambridge, 2000), 1-17, Springer, Berlin, 2002.
  9. M. Bourdon: Immeubles hyperboliques, dimension conforme et rigidite de Mostow. (French) [Hyperbolic buildings, conformal dimension and Mostow rigidity]. Geom. Funct. Anal. 7 (1997), no. 2, 245-268. https://doi.org/10.1007/PL00001619
  10. B. Colbois, C. Vernicos & P. Verovic: L'aire des triangles ideaux en geometrie de Hilbert (French) [The area of ideal triangles in Hilbert geometry]. Enseign. Math. (2) 50 (2004), no. 3-4, 203-237.
  11. C. Connell & B. Farb: Minimal entropy rigidity for lattices in products of rank one symmetric spaces. Comm. Anal. Geom. 11 (2003), no. 5, 1001-1026. https://doi.org/10.4310/CAG.2003.v11.n5.a7
  12. C. Connell & B. Farb:Some recent applications of the barycenter method in geometry. Topology and Geometry of Manifolds, Proc. Symp. in Pure Math., G. Matic and C. McCrory, Eds., 2003, 19-51.
  13. F. Dalbo, M. Peigne, J. Picaud & A. Sambusetti: On the growth of nonuniform lattices in pinched negatively curved manifolds. J. Reine Angew. Math. 627 (2009), 31-52.
  14. D. Egloff: Uniform Finsler Hadamard manifolds. Ann. Inst. H. Poincare Phys. Theor. 66 (1997), no. 3, 323-357.
  15. A. Eskin & M. Mirzakhani: Counting closed geodesics in moduli space. J. Mod. Dyn. 5 (2011), no. 1, 71-105, https://doi.org/10.3934/jmd.2011.5.71
  16. B. Farb, C. Hruska & A. Thomas: Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices. In "Geometry, Rigidity, and Group Actions", Benson Farb and David Fisher (eds), The University of Chicago Press, Chicago 2011.
  17. A. Katok: Entropy and closed geodesics. Ergod. Th. Dyn. Sys. 2 (1982), 339-365.
  18. I. Kapovich & T. Nagnibeda: The Patterson-Sullivan embedding and minimal volume entropy for outer space. Geom. Funct. Anal. (GAFA) 17 (2007), no 4, 1201-1236, preprint (http://arxiv.org/abs/math.GR/0504445), 2005. https://doi.org/10.1007/s00039-007-0621-z
  19. F. Ledrappier: Harmonic measures and BowenMargulis measures. Israel J. of Math. 71 (1990), no. 3, 275-287. https://doi.org/10.1007/BF02773746
  20. F. Ledrappier & S. Lim: Volume entropy of hyperbolic buildings. J. Mod. Dyn. 4 (2010), no.1, 139-165. https://doi.org/10.3934/jmd.2010.4.139
  21. F. Ledrappier & X.Wang: An integral formula for the volume entropy with applications to rigidity. J. Differential Geom. 85, (2010) no. 3, 461-477. https://doi.org/10.4310/jdg/1292940691
  22. E. Leuzinger: Isoperimetric inequalities and random walks on quotients of graphs and buildings. Math. Z. 248 (2004), no. 1, 101-112.
  23. E. Leuzinger: Entropy of the geodesic flow for metric spaces and Bruhat-Tits buildings. Adv. Geom. 6 (2006), no. 3, 475-491. https://doi.org/10.1515/ADVGEOM.2006.029
  24. Z. Lian & L.-S. Young: Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces. Ann. Henri Poincare 12 (2011), no. 6, 1081-1108. https://doi.org/10.1007/s00023-011-0100-9
  25. S. Lim: Minimal volume entropy for graphs. Trans. Amer. Math. Soc. 360 (2008), 5089-5100. https://doi.org/10.1090/S0002-9947-08-04227-X
  26. E. Makover & J. McGowan: The length of closed geodesics on random Riemann surfaces. Geom. Dedicata 151 (2011), 207-220. https://doi.org/10.1007/s10711-010-9528-1
  27. A. Manning: A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergodic Theory Dynamical Systems 1 (1981), no. 4, 451-459.
  28. A. Manning: Topological entropy for geodesic flows. Ann. of Math. (2) 110 (1979), no. 3, 567-573. https://doi.org/10.2307/1971239
  29. G. Margulis: On Some Aspects of the Theory of Anosov Systems. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Monographs in Mathematics, Springer- Verlag, Berlin, 2004.
  30. I. Rivin: Walks on Free groups and other stories, twelve years later. preprint, 2011.
  31. T. Roblin: Ergodicite et equidistribution en courbure negative. Mem. Soc. Math. Fr. (N.S.), (95):vi+96, 2003.
  32. Z. Shen: Lectures on Finsler geometry. World Scientific. xiv, 2001.
  33. L.-S. Young: Dimension, entropy and Lyapunov exponents. Ergodic Theory Dynamical Systems 2 (1982), no. 1, 109-124. https://doi.org/10.1017/S0143385700009615
  34. C. Vernicos: Lipschitz characterization of convex polytopal Hilbert geometries. preprint, 2008.
  35. D. Volchenkov & P. Blanchard: Transport networks revisited: why dual graphs. preprint, 2007.