Losing Hausdorff dimension while generating pseudogroups

Paweł Walczak

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 3, page 211-237
  • ISSN: 0016-2736

Abstract

top
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 d i m H G of G. Since d i m H G ≤ d i m H X , the dimension loss d l H G = d i m H X - d i m H G can be considered as a “topological price” one has to pay to generate G. We collect some properties of d i m H and d l H (for example, both of them are invariant under Lipschitz isomorphisms of pseudogroups) and we either estimate or calculate d i m H G for pseudogroups arising from classical dynamical systems, group actions, foliations, etc.

How to cite

top

Walczak, Paweł. "Losing Hausdorff dimension while generating pseudogroups." Fundamenta Mathematicae 149.3 (1996): 211-237. <http://eudml.org/doc/212120>.

@article{Walczak1996,
abstract = {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.},
author = {Walczak, Paweł},
journal = {Fundamenta Mathematicae},
keywords = {pseudogroup; Hausdorff dimension; dimension loss; foliations},
language = {eng},
number = {3},
pages = {211-237},
title = {Losing Hausdorff dimension while generating pseudogroups},
url = {http://eudml.org/doc/212120},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Walczak, Paweł
TI - Losing Hausdorff dimension while generating pseudogroups
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 3
SP - 211
EP - 237
AB - 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.
LA - eng
KW - pseudogroup; Hausdorff dimension; dimension loss; foliations
UR - http://eudml.org/doc/212120
ER -

References

top
  1. [Ba] M. F. Barnsley, Fractals Everywhere, Academic Press, Boston, 1993. 
  2. [Be] A. F. Beardon, Iteration of Rational Functions, Grad. Texts in Math., Springer, New York, 1991. 
  3. [Bi] A. Biś, Entropy of topological directions, preprint. Zbl0909.54014
  4. [Bo] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES 50 (1979), 11-26. Zbl0439.30032
  5. [CL] C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Birkhäuser, Boston, 1985. Zbl0568.57002
  6. [Ca] A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. 26 (1993), 489-516. Zbl0785.57009
  7. [C1] J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. Zbl0606.57003
  8. [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. 
  9. [CC] J. Cantwell and L. Conlon, Foliations and subshifts, Tôhoku Math. J. 40 (1988), 165-187. 
  10. [Ch] C. Champetier, Petite simplification dans les groupes hyperboliques, Ann. Fac. Sci. Toulouse 3 (1994), 161-221. 
  11. [CDP] M. Coornaert, T. Delzant et A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Math. 1441, Springer, Berlin, 1991. 
  12. [Ed] G. A. Edgar, Measure, Topology and Fractal Geometry, Undergrad. Texts in Math., Springer, New York, 1990. 
  13. [Eg1] S. Egashira, Expansion growth of foliations, Ann. Fac. Sci. Toulouse 2 (1993), 15-52. Zbl0789.58063
  14. [Eg2] S. Egashira, Expansion growth of horospherical foliations, J. Fac. Sci. Univ. Tokyo 40 (1994), 663-682. Zbl0813.57024
  15. [Fa] K. Falconer, Fractal Geometry, Wiley, Chichester, 1990. 
  16. [FM] K. Falconer and D. Marsh, Classification of quasi-circles by Hausdorff dimension, Nonlinearity 2 (1989), 489-493. Zbl0684.58023
  17. [Fr] H. Frings, Generalized entropy for foliations, Thesis, Düsseldorf, 1991. Zbl0748.57012
  18. [Ga] F. R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959. 
  19. [Gar] L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983), 285-311. Zbl0524.58026
  20. [Gh] E. Ghys, Rigidité différentiable des groupes fuchsiens, Publ. Math. IHES, to appear. 
  21. [GH1] E. Ghys et P. de la Harpe (eds.), Sur les Groupes Hyperboliques d'après Mikhael Gromov, Birkhäuser, Boston, 1990. Zbl0731.20025
  22. [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. 
  23. [GHV] E. Ghys, P. de la Harpe and A. Verjovsky (eds.), Group Theory from Geometrical Viewpoint, World. Sci., Singapore, 1991. 
  24. [GLW] E. Ghys, R. Langevin and P. Walczak, Entropie géométrique des feuilletages, Acta Math. 160 (1988), 105-142. 
  25. [Go] C. Godbillon, Feuilletages, Birkhäuser, Basel, 1991. 
  26. [Gr] M. Gromov, Hyperbolic groups, in: Essays in Group Theory, S. M. Gersten (ed.), MSRI Publ. 8, Springer, Berlin, 1987, 75-263. 
  27. [GLP] M. Gromov, J. Lafontaine et P. Pansu, Structures métriques sur les variétés riemanniennes, Cedic/Fernand Nathan, Paris, 1981. 
  28. [Ha] A. Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geom. 15 (1980), 269-284. Zbl0444.57016
  29. [HH] G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Parts A and B, Vieweg, Braunschweig, 1981 and 1983. 
  30. [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. 
  31. [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. 
  32. [Hu2] S. Hurder, Exceptional minimal sets of C 1 + α -group actions on the circle, Ergodic Theory Dynam. Systems 11 (1991), 455-467. 
  33. [Hu3] S. Hurder, Rigidity for Anosov actions of higher rank lattices, Ann. of Math. 135 (1992), 361-410. Zbl0754.58029
  34. [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. 
  35. [HK] S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Ann. of Math. 126 (1987), 221-275. Zbl0645.57021
  36. [In] T. Inaba, Examples of exceptional minimal sets, in: A Fête of Topology, Academic Press, Boston 1988, 95-100. 
  37. [IW] T. Inaba and P. Walczak, Transverse Hausdorff dimension of codim-1 C 2 -foliations, this issue, 239-244. Zbl0860.57021
  38. [Ke] M. Kellum, Uniformly quasi-isometric foliations, Ergodic Theory Dynam. Systems 13 (1993), 101-122. Zbl0784.58060
  39. [Kl] W. Klingenberg, Riemannian Geometry, de Gruyter, Berlin, 1982. 
  40. [LW1] R. Langevin and P. Walczak, Entropie d'une dynamique, C. R. Acad. Sci. Paris 312 (1991), 141-144. Zbl0723.54020
  41. [LW2] R. Langevin and P. Walczak, Entropy, transverse entropy and partitions of unity, Ergodic Theory Dynam. Systems 14 (1994), 551-563. Zbl0856.54025
  42. [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. 
  43. [Le] G. Levitt, On the cost of generating an equivalence relation, Ergodic Theory Dynam. Systems 15 (1995), 1173-1181. Zbl0843.28010
  44. [Ma] J. Marion, Mesure de Hausdorff d'un fractal à similitude interne, Ann. Sci. Math. Québec 10 (1986), 51-84. Zbl0613.28007
  45. [Pa] W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55-65. Zbl0127.35301
  46. [Pl] J. Plante, Foliations with measure preserving holonomy, Ann. of Math. 102 (1975), 327-361. Zbl0314.57018
  47. [Ru] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. 
  48. [Sm] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-815. 
  49. [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. 
  50. [Ta] I. Tamura, Topology of Foliations: An Introduction, Amer. Math. Soc., Providence, 1992. 
  51. [TT] S. J. Taylor and C. Tricot, Packing measure and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679-699. Zbl0537.28003
  52. [W1] P. Walczak, Dynamics of the geodesic flow of a foliation, Ergodic Theory Dynam. Systems 8 (1988), 637-650. Zbl0663.53028
  53. [W2] P. Walczak, Existence of smooth invariant measures for geodesic flows of foliations of Riemannian manifolds, Proc. Amer. Math. Soc. 120 (1994), 903-906. Zbl0804.53045
  54. [Wa] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. 
  55. [Wi] B. Wirtz, Entropies, Thesis, Dijon, 1993. 
  56. [Zi1] R. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. 112 (1980), 511-529. Zbl0468.22011
  57. [Zi2] R. Zimmer, Ergodic theory, semisimple Lie groups and foliations by manifolds of negative curvature, Publ. Math. IHES 55 (1982). 
  58. [Zi3] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.