Transverse Hausdorff dimension of codim-1 C2-foliations

Takashi Inaba; Paweł Walczak

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 3, page 239-244
  • ISSN: 0016-2736

Abstract

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The Hausdorff dimension of the holonomy pseudogroup of a codimension-one foliation ℱ is shown to coincide with the Hausdorff dimension of the space of compact leaves (traced on a complete transversal) when ℱ is non-minimal, and to be equal to zero when ℱ is minimal with non-trivial leaf holonomy.

How to cite

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Inaba, Takashi, and Walczak, Paweł. "Transverse Hausdorff dimension of codim-1 C2-foliations." Fundamenta Mathematicae 149.3 (1996): 239-244. <http://eudml.org/doc/212121>.

@article{Inaba1996,
abstract = {The Hausdorff dimension of the holonomy pseudogroup of a codimension-one foliation ℱ is shown to coincide with the Hausdorff dimension of the space of compact leaves (traced on a complete transversal) when ℱ is non-minimal, and to be equal to zero when ℱ is minimal with non-trivial leaf holonomy.},
author = {Inaba, Takashi, Walczak, Paweł},
journal = {Fundamenta Mathematicae},
keywords = {Hausdorff dimension; holonomy pseudogroup; foliation; compact leaves},
language = {eng},
number = {3},
pages = {239-244},
title = {Transverse Hausdorff dimension of codim-1 C2-foliations},
url = {http://eudml.org/doc/212121},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Inaba, Takashi
AU - Walczak, Paweł
TI - Transverse Hausdorff dimension of codim-1 C2-foliations
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 3
SP - 239
EP - 244
AB - The Hausdorff dimension of the holonomy pseudogroup of a codimension-one foliation ℱ is shown to coincide with the Hausdorff dimension of the space of compact leaves (traced on a complete transversal) when ℱ is non-minimal, and to be equal to zero when ℱ is minimal with non-trivial leaf holonomy.
LA - eng
KW - Hausdorff dimension; holonomy pseudogroup; foliation; compact leaves
UR - http://eudml.org/doc/212121
ER -

References

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  1. [Ar] V. I. Arnold, Small denominators. I. Mappings of the circumference onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1962), 21-86 (in Russian); English transl.: Amer. Math. Soc. Transl. 46 (1965), 213-284. 
  2. [Ed] G. A. Edgar, Measure, Topology and Fractal Geometry, Undergrad. Texts in Math., Springer, New York, 1990. 
  3. [HH] G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Part B, Vieweg, Braunschweig, 1983. 
  4. [He] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES 49 (1979), 1-233. Zbl0448.58019
  5. [La] C. Lamoureux, Quelques conditions d'existence de feuilles compactes, Ann. Inst. Fourier (Grenoble) 24 (4) (1974), 229-240. Zbl0287.57009
  6. [Sa] R. Sacksteder, Foliations and pseudogroups, Amer. J. Math. 87 (1965), 79-102. Zbl0136.20903
  7. [Ta] I. Tamura, Topology of Foliations: An Introduction, Amer. Math. Soc., Providence, 1992. 
  8. [Wa] P. Walczak, Losing Hausdorff dimension while generating pseudogroups, this issue, 211-237. Zbl0861.54033

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