Quasi-symmetric invariant properties of Cantor metric spaces
Annales de l'Institut Fourier, Volume 69 (2019) no. 6, p. 2681-2721

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.

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.

Received : 2018-03-16
Revised : 2018-11-17
Accepted : 2019-01-17
Published online : 2019-10-29
DOI : https://doi.org/10.5802/aif.3305
Classification:  54E40,  54F45
Keywords: Cantor metric space, Quasi-symmetric invariant
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     author = {Ishiki, Yoshito},
     title = {Quasi-symmetric invariant properties of Cantor metric spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {6},
     year = {2019},
     pages = {2681-2721},
     doi = {10.5802/aif.3305},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_6_2681_0}
}
Ishiki, Yoshito. Quasi-symmetric invariant properties of Cantor metric spaces. Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2681-2721. doi : 10.5802/aif.3305. https://aif.centre-mersenne.org/item/AIF_2019__69_6_2681_0/

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