The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables

Zelený, Miroslav

doi:10.5802/aif.2356

Zelený, Miroslav

Annales de l'Institut Fourier, Volume 58 (2008) no. 2, p. 405-428

Abstract
Résumé

We construct a differentiable function $f : R^{n} \to R$ ( $n \geq 2$ ) such that the set ${(\nabla f)}^{- 1} (B (0, 1))$ is a nonempty set of Hausdorff dimension $1$ . This answers a question posed by Z. Buczolich.

On construit une fonction différentiable $f : R^{n} \to R$ ( $n \geq 2$ ) telle que l’ensemble ${(\nabla f)}^{- 1} (B (0, 1))$ est non vide et sa dimension de Hausdorff est $1$ . C’est une réponse à une question posée par Z. Buczolich.

MR 2410378 | Zbl 1154.26016

DOI : https://doi.org/10.5802/aif.2356
Classification: 26B05, 28A75
Keywords: Denjoy–Clarkson property, gradient, Hausdorff measure, infinite game

@article{AIF_2008__58_2_405_0,
     author = {Zelen\'y, Miroslav},
     title = {The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     pages = {405-428},
     doi = {10.5802/aif.2356},
     mrnumber = {2410378},
     zbl = {1154.26016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_2_405_0}
}

Zelený, Miroslav. The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 405-428. doi : 10.5802/aif.2356. http://www.numdam.org/item/AIF_2008__58_2_405_0/

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