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1996 | 149 | 3 | 211-237
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Losing Hausdorff dimension while generating pseudogroups

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Considering different finite sets of maps generating a pseudogroup G of locally Lipschitz homeomorphisms between open subsets of a compact metric space X we arrive at a notion of a Hausdorff dimension $dim_H G$ of G. Since $dim_H G ≤ dim_H X$, the dimension loss $dl_HG = dim_HX - dim_H G$ can be considered as a "topological price" one has to pay to generate G. We collect some properties of $dim_H$ and $dl_H$ (for example, both of them are invariant under Lipschitz isomorphisms of pseudogroups) and we either estimate or calculate $dim_HG$ for pseudogroups arising from classical dynamical systems, group actions, foliations, etc.
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  • [Ba] M. F. Barnsley, Fractals Everywhere, Academic Press, Boston, 1993.
  • [Be] A. F. Beardon, Iteration of Rational Functions, Grad. Texts in Math., Springer, New York, 1991.
  • [Bi] A. Biś, Entropy of topological directions, preprint.
  • [Bo] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES 50 (1979), 11-26.
  • [CL] C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Birkhäuser, Boston, 1985.
  • [Ca] A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. 26 (1993), 489-516.
  • [C1] J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148.
  • [C2] J. W. Cannon, The theory of negatively curved spaces and groups, in: Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane and C. Series (eds.), Oxford Univ. Press, Oxford, 1991, 315-369.
  • [CC] J. Cantwell and L. Conlon, Foliations and subshifts, Tôhoku Math. J. 40 (1988), 165-187.
  • [Ch] C. Champetier, Petite simplification dans les groupes hyperboliques, Ann. Fac. Sci. Toulouse 3 (1994), 161-221.
  • [CDP] M. Coornaert, T. Delzant et A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Math. 1441, Springer, Berlin, 1991.
  • [Ed] G. A. Edgar, Measure, Topology and Fractal Geometry, Undergrad. Texts in Math., Springer, New York, 1990.
  • [Eg1] S. Egashira, Expansion growth of foliations, Ann. Fac. Sci. Toulouse 2 (1993), 15-52.
  • [Eg2] S. Egashira, Expansion growth of horospherical foliations, J. Fac. Sci. Univ. Tokyo 40 (1994), 663-682.
  • [Fa] K. Falconer, Fractal Geometry, Wiley, Chichester, 1990.
  • [FM] K. Falconer and D. Marsh, Classification of quasi-circles by Hausdorff dimension, Nonlinearity 2 (1989), 489-493.
  • [Fr] H. Frings, Generalized entropy for foliations, Thesis, Düsseldorf, 1991.
  • [Ga] F. R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959.
  • [Gar] L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983), 285-311.
  • [Gh] E. Ghys, Rigidité différentiable des groupes fuchsiens, Publ. Math. IHES, to appear.
  • [GH1] E. Ghys et P. de la Harpe (eds.), Sur les Groupes Hyperboliques d'après Mikhael Gromov, Birkhäuser, Boston, 1990.
  • [GH2] E. Ghys et P. de la Harpe (eds.), Infinite groups as geometric objects (after Gromov), in: Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane and C. Series (eds.), Oxford Univ. Press, Oxford, 1991, 299-314.
  • [GHV] E. Ghys, P. de la Harpe and A. Verjovsky (eds.), Group Theory from Geometrical Viewpoint, World. Sci., Singapore, 1991.
  • [GLW] E. Ghys, R. Langevin and P. Walczak, Entropie géométrique des feuilletages, Acta Math. 160 (1988), 105-142.
  • [Go] C. Godbillon, Feuilletages, Birkhäuser, Basel, 1991.
  • [Gr] M. Gromov, Hyperbolic groups, in: Essays in Group Theory, S. M. Gersten (ed.), MSRI Publ. 8, Springer, Berlin, 1987, 75-263.
  • [GLP] M. Gromov, J. Lafontaine et P. Pansu, Structures métriques sur les variétés riemanniennes, Cedic/Fernand Nathan, Paris, 1981.
  • [Ha] A. Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geom. 15 (1980), 269-284.
  • [HH] G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Parts A and B, Vieweg, Braunschweig, 1981 and 1983.
  • [Hi] M. Hirsch, A stable analytic foliation with only exceptional minimal sets, in: Dynamical Systems, Warwick 1974, Lecture Notes in Math. 468, Springer, Berlin, 1975, 9-10.
  • [Hu1] S. Hurder, Ergodic theory of foliations and a theorem of Sacksteder, in: Dynamical Systems, Proc. Special Year at Univ. of Maryland, 1986/87, Lecture Notes in Math. 1342, Springer, Berlin, 1988, 291-328.
  • [Hu2] S. Hurder, Exceptional minimal sets of $C^{1+α}$-group actions on the circle, Ergodic Theory Dynam. Systems 11 (1991), 455-467.
  • [Hu3] S. Hurder, Rigidity for Anosov actions of higher rank lattices, Ann. of Math. 135 (1992), 361-410.
  • [Hu4] S. Hurder, A survey on rigidity theory of Anosov actions, in: Differential Topology, Foliations and Group Actions, Rio de Janeiro 1992, P. Schweitzer et al. (eds.), Contemp. Math. 161, Amer. Math. Soc., Providence, 1994, 143-173.
  • [HK] S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Ann. of Math. 126 (1987), 221-275.
  • [In] T. Inaba, Examples of exceptional minimal sets, in: A Fête of Topology, Academic Press, Boston 1988, 95-100.
  • [IW] T. Inaba and P. Walczak, Transverse Hausdorff dimension of codim-1 $C^2$-foliations, this issue, 239-244.
  • [Ke] M. Kellum, Uniformly quasi-isometric foliations, Ergodic Theory Dynam. Systems 13 (1993), 101-122.
  • [Kl] W. Klingenberg, Riemannian Geometry, de Gruyter, Berlin, 1982.
  • [LW1] R. Langevin and P. Walczak, Entropie d'une dynamique, C. R. Acad. Sci. Paris 312 (1991), 141-144.
  • [LW2] R. Langevin and P. Walczak, Entropy, transverse entropy and partitions of unity, Ergodic Theory Dynam. Systems 14 (1994), 551-563.
  • [LW3] R. Langevin and P. Walczak, Some invariants measuring dynamics of codimension-one foliations, in: Geometric Study of Foliations, T. Mizutani et al. (eds.), World Sci., Singapore, 1994, 345-358.
  • [Le] G. Levitt, On the cost of generating an equivalence relation, Ergodic Theory Dynam. Systems 15 (1995), 1173-1181.
  • [Ma] J. Marion, Mesure de Hausdorff d'un fractal à similitude interne, Ann. Sci. Math. Québec 10 (1986), 51-84.
  • [Pa] W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55-65.
  • [Pl] J. Plante, Foliations with measure preserving holonomy, Ann. of Math. 102 (1975), 327-361.
  • [Ru] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.
  • [Sm] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-815.
  • [Su] D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, in: Mathematics into the 21st Century, Vol. 2, Amer. Math. Soc. Cent. Publ., Providence, 1991, 417-466.
  • [Ta] I. Tamura, Topology of Foliations: An Introduction, Amer. Math. Soc., Providence, 1992.
  • [TT] S. J. Taylor and C. Tricot, Packing measure and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679-699.
  • [W1] P. Walczak, Dynamics of the geodesic flow of a foliation, Ergodic Theory Dynam. Systems 8 (1988), 637-650.
  • [W2] P. Walczak, Existence of smooth invariant measures for geodesic flows of foliations of Riemannian manifolds, Proc. Amer. Math. Soc. 120 (1994), 903-906.
  • [Wa] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.
  • [Wi] B. Wirtz, Entropies, Thesis, Dijon, 1993.
  • [Zi1] R. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. 112 (1980), 511-529.
  • [Zi2] R. Zimmer, Ergodic theory, semisimple Lie groups and foliations by manifolds of negative curvature, Publ. Math. IHES 55 (1982).
  • [Zi3] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984.
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