no. 6
Quasi-symmetric invariant properties of Cantor metric spaces
[Propriétés invariantes quasi-symmétriques des espaces métriques de Cantor]
Ishiki, Yoshito1
1 University of Tsukuba Graduate School of Pure and Applied Sciences Tennodai 1-1-1 Tsukuba, Ibaraki, 305-8571 (Japan)
Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2681-2721.

Pour les espaces métriques, la propriété de doublage, la déconnexion uniforme et la perfection uniforme sont connues comme des propriétés invariantes par les quasi-symétries. Le théorème d’uniformisation de David–Semmes stipule que si un espace métrique compact satisfait toutes ces trois propriétés, il est quasi-symétriquement équivalent à l’ensemble triadique de Cantor. Nous disons qu’un espace métrique de Cantor est standard s’il satisfait toutes les trois propriétés, et exotique. Sinon, dans cet article, nous concluons que pour chaque type exotique la classe de tous les jauges conformales des espaces métriques de Cantor a exactement la cardinalité du continuum. En tant que sous-produit de notre étude, nous avons montré qu’il existe un espace métrique de Cantor ayant la dimension de Hausdorff et la dimension d’Assouad prescrites.

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David–Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this paper, we conclude that for each of exotic type the class of all the conformal gauges of Cantor metric spaces exactly has continuum cardinality. As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assouad dimension.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3305
Classification : 54E40, 54F45
Keywords: Cantor metric space, Quasi-symmetric invariant
Mot clés : Espace métrique du Cantor, Invariants Quasi-symétrique
Ishiki, Yoshito 1

1 University of Tsukuba Graduate School of Pure and Applied Sciences Tennodai 1-1-1 Tsukuba, Ibaraki, 305-8571 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2019__69_6_2681_0,
     author = {Ishiki, Yoshito},
     title = {Quasi-symmetric invariant properties of {Cantor} metric spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {2681--2721},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {6},
     year = {2019},
     doi = {10.5802/aif.3305},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3305/}
}
TY  - JOUR
AU  - Ishiki, Yoshito
TI  - Quasi-symmetric invariant properties of Cantor metric spaces
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 2681
EP  - 2721
VL  - 69
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3305/
DO  - 10.5802/aif.3305
LA  - en
ID  - AIF_2019__69_6_2681_0
ER  - 
%0 Journal Article
%A Ishiki, Yoshito
%T Quasi-symmetric invariant properties of Cantor metric spaces
%J Annales de l'Institut Fourier
%D 2019
%P 2681-2721
%V 69
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3305/
%R 10.5802/aif.3305
%G en
%F AIF_2019__69_6_2681_0
Ishiki, Yoshito. Quasi-symmetric invariant properties of Cantor metric spaces. Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2681-2721. doi : 10.5802/aif.3305. https://aif.centre-mersenne.org/articles/10.5802/aif.3305/

[1] Brouwer, Luitzen E. J. On the structure of perfect sets of points, Proc. Akad. Amsterdam, Volume 12 (1910), pp. 785-794

[2] David, Guy; Semmes, Stephen Fractured fractals and broken dreams: Self similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications, 7, Oxford University Press, 1997 | Zbl

[3] Heinonen, Juha Lectures on analysis on metric spaces, Universitext, Springer, 2001 | DOI | Zbl

[4] Kechris, Alexander S. Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer, 1994 | Zbl

[5] Kuratowski, Kazimierz Topology. Vol. I, Academic Press Inc., 1966 | Zbl

[6] Luosto, Kerkko Ultrametric spaces bi-Lipschitz embeddable in ℝ n , Fundam. Math., Volume 150 (1996) no. 1, pp. 25-42 | MR | Zbl

[7] Mackay, John M.; Tyson, Jeremy T. Conformal dimension: theory and application, University Lecture Series, 54, American Mathematical Society, 2010 | Zbl

[8] Semmes, Stephen Metric spaces and mapping seen at many scales, Metric structures for Riemannian and non-Riemannian Spaces (Progress in Mathematics), Volume 152 (1999) (Appendix) | Zbl

[9] Semmes, Stephen Cellular structures, quasisymmetric mappings and spaces of homogeneous type (2007) (https://arxiv.org/abs/0711.1333v1)

[10] Semmes, Stephen An introduction to the geometry of ultrametric spaces (2007) (https://arxiv.org/abs/0711.0709v1)

[11] Urysohn, Pavel Mémoire sur les multiplicités cantoriennes, Fundam. Math., Volume 7 (1925), pp. 30-137 | DOI | Zbl

[12] Willard, Stephen General topology, Dover Publications, 2004 (Reprint of the 1970 original) | MR | Zbl

Cité par Sources :